1.8.1 The Limit Laws Suppose lim x!a f(x) = L and lim x!a g(x) = M 1.lim x!a f g = L M The limit of a sum or di erence of functions is the sum or di erence of their limit values. 2.lim x!a cf(x) = cL = clim x!a f(x) The constant can be pulled out of a limit. 3.lim x!a f(x) g(x) = lim x!a f(x) lim x!a g(x) = LM Please note that you can only. Calculus 8th Edition answers to Chapter 1 - Functions and Limits - 1.8 Continuity - 1.8 Exercises - Page 91 2 including work step by step written by community members like you. Textbook Authors: Stewart, James, ISBN-10:, ISBN-13: 978-1-28574-062-1, Publisher: Cengage. MATH 1200: Calculus for Scientists I 1: Limit and Continuity of Functions. 1.8: Limits and continuity of Inverse Trigonometric functions Last updated; Save as PDF. Calculus sequences-and-series limits. None of the remaining terms will be negative, so you know this is a bottom limit on what you are going to get.

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Section 2-7 : Limits at Infinity, Part I

In the previous section we saw limits that were infinity and it’s now time to take a look at limits at infinity. By limits at infinity we mean one of the following two limits.

[mathop {lim }limits_{x to infty } fleft( x right)hspace{0.25in}hspace{0.25in}hspace{0.25in}mathop {lim }limits_{x to - infty } fleft( x right)]

In other words, we are going to be looking at what happens to a function if we let (x) get very large in either the positive or negative sense. Also, as we’ll soon see, these limits may also have infinity as a value.

First, let’s note that the set of Facts from the Infinite Limit section also hold if we replace the (mathop {lim }limits_{x to ,c} ) with (mathop {lim }limits_{x to infty } ) or (mathop {lim }limits_{x to - infty } ). The proof of this is nearly identical to the proof of the original set of facts with only minor modifications to handle the change in the limit and so is left to you. We won’t need these facts much over the next couple of sections but they will be required on occasion.

In fact, many of the limits that we’re going to be looking at we will need the following two facts.

Fact 1

1. If (r) is a positive rational number and (c) is any real number then, [mathop {lim }limits_{x to infty } frac{c}{{{x^r}}} = 0]
2. If (r) is a positive rational number, (c) is any real number and ({x^r}) is defined for (x < 0) then, [mathop {lim }limits_{x to - infty } frac{c}{{{x^r}}} = 0]

The first part of this fact should make sense if you think about it. Because we are requiring (r > 0) we know that (x^{r}) will stay in the denominator. Next as we increase (x) then (x^{r}) will also increase. So, we have a constant divided by an increasingly large number and so the result will be increasingly small. Or, in the limit we will get zero.

The second part is nearly identical except we need to worry about (x^{r}) being defined for negative (x). This condition is here to avoid cases such as (r = frac{1}{2}). If this (r) were allowed we’d be taking the square root of negative numbers which would be complex and we want to avoid that at this level.

Note as well that the sign of (c) will not affect the answer. Regardless of the sign of (c) we’ll still have a constant divided by a very large number which will result in a very small number and the larger (x) get the smaller the fraction gets. The sign of (c) will affect which direction the fraction approaches zero (i.e. from the positive or negative side) but it still approaches zero.

If you think about it this is really a special case of the last Fact from the Facts in the previous section. However, to see a direct proof of this fact see the Proof of Various Limit Properties section in the Extras chapter.

Let’s start off the examples with one that will lead us to a nice idea that we’ll use on a regular basis about limits at infinity for polynomials.

Example 1 Evaluate each of the following limits.

1. (mathop {lim }limits_{x to infty } left( {2{x^4} - {x^2} - 8x} right))
2. (mathop {lim }limits_{t to - infty } left( {{textstyle{1 over 3}}{t^5} + 2{t^3} - {t^2} + 8} right))

Show All SolutionsHide All Solutionsa (mathop {lim }limits_{x to infty } left( {2{x^4} - {x^2} - 8x} right)) Show Solution

Our first thought here is probably to just “plug” infinity into the polynomial and “evaluate” each term to determine the value of the limit. It is pretty simple to see what each term will do in the limit and so this seems like an obvious step, especially since we’ve been doing that for other limits in previous sections.

So, let’s see what we get if we do that. As (x) approaches infinity, then (x) to a power can only get larger and the coefficient on each term (the first and third) will only make the term even larger. So, if we look at what each term is doing in the limit we get the following,

[mathop {lim }limits_{x to infty } left( {2{x^4} - {x^2} - 8x} right) = infty - infty - infty ]

Now, we’ve got a small, but easily fixed, problem to deal with. We are probably tempted to say that the answer is zero (because we have an infinity minus an infinity) or maybe ( - infty )(because we’re subtracting two infinities off of one infinity). However, in both cases we’d be wrong. This is one of those indeterminate forms that we first started seeing in a previous section.

Infinities just don’t always behave as real numbers do when it comes to arithmetic. Without more work there is simply no way to know what (infty - infty ) will be and so we really need to be careful with this kind of problem. To read a little more about this see the Types of Infinity section in the Extras chapter.

So, we need a way to get around this problem. What we’ll do here is factor the largest power of (x) out of the whole polynomial as follows,

[mathop {lim }limits_{x to infty } left( {2{x^4} - {x^2} - 8x} right) = mathop {lim }limits_{x to infty } left[ {{x^4}left( {2 - frac{1}{{{x^2}}} - frac{8}{{{x^3}}}} right)} right]]

If you’re not sure you agree with the factoring above (there’s a chance you haven’t really been asked to do this kind of factoring prior to this) then recall that to check all you need to do is multiply the ({x^4}) back through the parenthesis to verify it was done correctly. Also, an easy way to remember how to do this kind of factoring is to note that the second term is just the original polynomial divided by ({x^4}). This will always work when factoring a power of (x) out of a polynomial.

Now for each of the terms we have,

[mathop {lim }limits_{x to infty } {x^4} = infty hspace{0.25in}hspace{0.25in}hspace{0.25in}mathop {lim }limits_{x to infty } left( {2 - frac{1}{{{x^2}}} - frac{8}{{{x^3}}}} right) = 2]

The first limit is clearly infinity and for the second limit we’ll use the fact above on the last two terms. Therefore using Fact 2 from the previous section we see value of the limit will be,

[mathop {lim }limits_{x to infty } left( {2{x^4} - {x^2} - 8x} right) = infty ]
b (mathop {lim }limits_{t to - infty } left( {{textstyle{1 over 3}}{t^5} + 2{t^3} - {t^2} + 8} right)) Show Solution

We’ll work this part much quicker than the previous part. All we need to do is factor out the largest power of (t) to get the following,

[mathop {lim }limits_{t to - infty } left( {{textstyle{1 over 3}}{t^5} + 2{t^3} - {t^2} + 8} right) = mathop {lim }limits_{t to - infty } left[ {{t^5}left( {frac{1}{3} + frac{2}{{{t^2}}} - frac{1}{{{t^3}}} + frac{8}{{{t^5}}}} right)} right]]

Remember that all you need to do to get the factoring correct is divide the original polynomial by the power of (t) we’re factoring out, ({t^5}) in this case.

Now all we need to do is take the limit of the two terms. In the first don’t forget that since we’re going out towards ( - infty ) and we’re raising (t) to the 5^th power that the limit will be negative (negative number raised to an odd power is still negative). In the second term we’ll again make heavy use of the fact above to see that is a finite number.

Therefore, using a modification of the Facts from the previous section the value of the limit is,

[mathop {lim }limits_{t to - infty } left( {{textstyle{1 over 3}}{t^5} + 2{t^3} - {t^2} + 8} right) = - infty ]

Okay, now that we’ve seen how a couple of polynomials work we can give a simple fact about polynomials in general.

Fact 2

If (pleft( x right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + cdots + {a_1}x + {a_0}) is a polynomial of degree (n) (i.e. ({a_n} ne 0)) then,

[mathop {lim }limits_{x to infty } pleft( x right) = mathop {lim }limits_{x to infty } {a_n}{x^n}hspace{0.25in}hspace{0.25in}hspace{0.25in}mathop {lim }limits_{x to - infty } pleft( x right) = mathop {lim }limits_{x to - infty } {a_n}{x^n}]

What this fact is really saying is that when we take a limit at infinity for a polynomial all we need to really do is look at the term with the largest power and ask what that term is doing in the limit since the polynomial will have the same behavior.

You can see the proof in the Proof of Various Limit Properties section in the Extras chapter.

Let’s now move into some more complicated limits.

Example 2 Evaluate both of the following limits. [mathop {lim }limits_{x to infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}}hspace{0.75in}mathop {lim }limits_{x to - infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}}] Show Solution

First, the only difference between these two is that one is going to positive infinity and the other is going to negative infinity. Sometimes this small difference will affect the value of the limit and at other times it won’t.

Let’s start with the first limit and as with our first set of examples it might be tempting to just “plug” in the infinity. Since both the numerator and denominator are polynomials we can use the above fact to determine the behavior of each. Doing this gives,

[mathop {lim }limits_{x to infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}} = frac{infty }{{ - infty }}]

This is yet another indeterminateform. In this case we might be tempted to say that the limit is infinity (because of the infinity in the numerator), zero (because of the infinity in the denominator) or -1 (because something divided by itself is one). There are three separate arithmetic “rules” at work here and without work there is no way to know which “rule” will be correct and to make matters worse it’s possible that none of them may work and we might get a completely different answer, say ( - frac{2}{5}) to pick a number completely at random.

So, when we have a polynomial divided by a polynomial we’re going to proceed much as we did with only polynomials. We first identify the largest power of (x) in the denominator (and yes, we only look at the denominator for this) and we then factor this out of both the numerator and denominator. Doing this for the first limit gives,

[mathop {lim }limits_{x to infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}} = mathop {lim }limits_{x to infty } frac{{{x^4}left( {2 - frac{1}{{{x^2}}} + frac{8}{{{x^3}}}} right)}}{{{x^4}left( { - 5 + frac{7}{{{x^4}}}} right)}}]

Once we’ve done this we can cancel the ({x^4}) from both the numerator and the denominator and then use the Fact 1 above to take the limit of all the remaining terms. This gives,

[begin{align*}mathop {lim }limits_{x to infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}} & = mathop {lim }limits_{x to infty } frac{{2 - frac{1}{{{x^2}}} + frac{8}{{{x^3}}}}}{{ - 5 + frac{7}{{{x^4}}}}} & = frac{{2 + 0 + 0}}{{ - 5 + 0}} & = - frac{2}{5}end{align*}]

In this case the indeterminate form was neither of the “obvious” choices of infinity, zero, or -1 so be careful with make these kinds of assumptions with this kind of indeterminate forms.

The second limit is done in a similar fashion. Notice however, that nowhere in the work for the first limit did we actually use the fact that the limit was going to plus infinity. In this case it doesn’t matter which infinity we are going towards we will get the same value for the limit.

[mathop {lim }limits_{x to - infty } frac{{2{x^4} - {x^2} + 8x}}{{ - 5{x^4} + 7}} = - frac{2}{5}]

In the previous example the infinity that we were using in the limit didn’t change the answer. This will not always be the case so don’t make the assumption that this will always be the case.

Let’s take a look at an example where we get different answers for each limit.

Example 3 Evaluate each of the following limits. [mathop {lim }limits_{x to infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}}hspace{0.75in}mathop {lim }limits_{x to - infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}}] Show Solution

The square root in this problem won’t change our work, but it will make the work a little messier.

Let’s start with the first limit. In this case the largest power of (x) in the denominator is just an (x). So, we need to factor an (x) out of the numerator and the denominator. When we are done factoring the (x) out we will need an (x) in both of the numerator and the denominator. To get this in the numerator we will have to factor an (x^{2}) out of the square root so that after we take the square root we will get an (x).

This is probably not something you’re used to doing, but just remember that when it comes out of the square root it needs to be an (x) and the only way have an (x) come out of a square root is to take the square root of (x^{2}) and so that is what we’ll need to factor out of the term under the radical. Here’s the factoring work for this part,

[begin{align*}mathop {lim }limits_{x to infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}} & = mathop {lim }limits_{x to infty } frac{{sqrt {{x^2}left( {3 + frac{6}{{{x^2}}}} right)} }}{{xleft( {frac{5}{x} - 2} right)}} & = mathop {lim }limits_{x to infty } frac{{sqrt {{x^2}} sqrt {3 + frac{6}{{{x^2}}}} }}{{xleft( {frac{5}{x} - 2} right)}}end{align*}]

This is where we need to be really careful with the square root in the problem. Don’t forget that

[sqrt {{x^2}} = left| x right|]

Square roots are ALWAYS positive and so we need the absolute value bars on the (x) to make sure that it will give a positive answer. This is not something that most people ever remember seeing in an Algebra class and in fact it’s not always given in an Algebra class. However, at this point it becomes absolutely vital that we know and use this fact. Using this fact the limit becomes,

[mathop {lim }limits_{x to infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}} = mathop {lim }limits_{x to infty } frac{{left| x right|sqrt {3 + frac{6}{{{x^2}}}} }}{{xleft( {frac{5}{x} - 2} right)}}]

Now, we can’t just cancel the (x)’s. We first will need to get rid of the absolute value bars. To do this let’s recall the definition of absolute value.

[left| x right| = left{ {begin{array}{rl}x & {{mbox{if }}x ge 0}{ - x} & {{mbox{if }}x < 0}end{array}} right.]

In this case we are going out to plus infinity so we can safely assume that the (x) will be positive and so we can just drop the absolute value bars. The limit is then,

[begin{align*}mathop {lim }limits_{x to infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}} & = mathop {lim }limits_{x to infty } frac{{xsqrt {3 + frac{6}{{{x^2}}}} }}{{xleft( {frac{5}{x} - 2} right)}} & = mathop {lim }limits_{x to infty } frac{{sqrt {3 + frac{6}{{{x^2}}}} }}{{frac{5}{x} - 2}} = frac{{sqrt {3 + 0} }}{{0 - 2}} = - frac{{sqrt 3 }}{2}end{align*}]

Let’s now take a look at the second limit (the one with negative infinity). In this case we will need to pay attention to the limit that we are using. The initial work will be the same up until we reach the following step.

[mathop {lim }limits_{x to - infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}} = mathop {lim }limits_{x to - infty } frac{{left| x right|sqrt {3 + frac{6}{{{x^2}}}} }}{{xleft( {frac{5}{x} - 2} right)}}]

In this limit we are going to minus infinity so in this case we can assume that (x) is negative. So, in order to drop the absolute value bars in this case we will need to tack on a minus sign as well. The limit is then,

[begin{align*}mathop {lim }limits_{x to - infty } frac{{sqrt {3{x^2} + 6} }}{{5 - 2x}} & = mathop {lim }limits_{x to - infty } frac{{ - xsqrt {3 + frac{6}{{{x^2}}}} }}{{xleft( {frac{5}{x} - 2} right)}} & = mathop {lim }limits_{x to - infty } frac{{ - sqrt {3 + frac{6}{{{x^2}}}} }}{{frac{5}{x} - 2}} & = frac{{sqrt 3 }}{2}end{align*}]

So, as we saw in the last two examples sometimes the infinity in the limit will affect the answer and other times it won’t. Note as well that it doesn’t always just change the sign of the number. It can on occasion completely change the value. We’ll see an example or two of this in the next section.

Before moving on to a couple of more examples let’s revisit the idea of asymptotes that we first saw in the previous section. Just as we can have vertical asymptotes defined in terms of limits we can also have horizontal asymptotes defined in terms of limits.

Definition

The function (f(x)) will have a horizontal asymptote at (y=L) if either of the following are true.

[mathop {lim }limits_{x to infty } fleft( x right) = Lhspace{0.25in}hspace{0.25in}hspace{0.25in}mathop {lim }limits_{x to - infty } fleft( x right) = L]

We’re not going to be doing much with asymptotes here, but it’s an easy fact to give and we can use the previous example to illustrate all the asymptote ideas we’ve seen in the both this section and the previous section. The function in the last example will have two horizontal asymptotes. It will also have a vertical asymptote. Here is a graph of the function showing these.

Let’s work another couple of examples involving rational expressions.

Example 4 Evaluate each of the following limits. [mathop {lim }limits_{z to infty } frac{{4{z^2} + {z^6}}}{{1 - 5{z^3}}}hspace{0.75in}mathop {lim }limits_{z to - infty } frac{{4{z^2} + {z^6}}}{{1 - 5{z^3}}}] Show Solution

Let’s do the first limit and in this case it looks like we will factor a (z^{3}) out of both the numerator and denominator. Remember that we only look at the denominator when determining the largest power of (z) here. There is a larger power of (z) in the numerator but we ignore it. We ONLY look at the denominator when doing this! So, doing the factoring gives,

[begin{align*}mathop {lim }limits_{z to infty } frac{{4{z^2} + {z^6}}}{{1 - 5{z^3}}} &= mathop {lim }limits_{z to infty } frac{{{z^3}left( {frac{4}{z} + {z^3}} right)}}{{{z^3}left( {frac{1}{{{z^3}}} - 5} right)}} & = mathop {lim }limits_{z to infty } frac{{frac{4}{z} + {z^3}}}{{frac{1}{{{z^3}}} - 5}}end{align*}]

When we take the limit we’ll need to be a little careful. The first term in the numerator and denominator will both be zero. However, the (z)³ in the numerator will be going to plus infinity in the limit and so the limit is,

[mathop {lim }limits_{z to infty } frac{{4{z^2} + {z^6}}}{{1 - 5{z^3}}} = frac{infty }{{ - 5}} = - infty ]

The final limit is negative because we have a quotient of positive quantity and a negative quantity.

Now, let’s take a look at the second limit. Note that the only different in the work is at the final “evaluation” step and so we’ll pick up the work there.

[mathop {lim }limits_{z to - infty } frac{{4{z^2} + {z^6}}}{{1 - 5{z^3}}} = mathop {lim }limits_{z to - infty } frac{{frac{4}{z} + {z^3}}}{{frac{1}{{{z^3}}} - 5}} = frac{{ - infty }}{{ - 5}} = infty ]

In this case the (z^{3}) in the numerator gives negative infinity in the limit since we are going out to minus infinity and the power is odd. The answer is positive since we have a quotient of two negative numbers.

Example 5 Evaluate the following limit. [mathop {lim }limits_{t to - infty } frac{{{t^2} - 5t - 9}}{{2{t^4} + 3{t^3}}}] Show Solution

In this case it looks like we will factor a ({t^4}) out of both the numerator and denominator. Doing this gives,

[begin{align*}mathop {lim }limits_{t to - infty } frac{{{t^2} - 5t - 9}}{{2{t^4} + 3{t^3}}} & = mathop {lim }limits_{t to - infty } frac{{{t^4}left( {frac{1}{{{t^2}}} - frac{5}{{{t^3}}} - frac{9}{{{t^4}}}} right)}}{{{t^4}left( {2 + frac{3}{t}} right)}} & = mathop {lim }limits_{t to - infty } frac{{frac{1}{{{t^2}}} - frac{5}{{{t^3}}} - frac{9}{{{t^4}}}}}{{2 + frac{3}{t}}} & = frac{0}{2} & = 0end{align*}]

In this case using Fact 1 we can see that the numerator is zero and so since the denominator is also not zero the fraction, and hence the limit, will be zero.

In this section we concentrated on limits at infinity with functions that only involved polynomials and/or rational expression involving polynomials. There are many more types of functions that we could use here. That is the subject of the next section.

To see a precise and mathematical definition of this kind of limit see the The Definition of the Limit section at the end of this chapter.

Given the sequence ({a_n} = {1/2^n} = 1/2, 1/4, 1/8, ldots), consider the following sums:

[begin{array}{ccccc}
a_1 &=& 1/2 &=& 1/2
a_1+a_2 &=& 1/2+1/4 &=& 3/4
a_1+a_2+a_3 &=& 1/2+1/4+1/8 &=& 7/8
a_1+a_2+a_3+a_4 &=& 1/2+1/4+1/8+1/16 & =& 15/16
end{array}]

In general, we can show that

[a_1+a_2+a_3+cdots +a_n = frac{2^n-1}{2^n} = 1-frac{1}{2^n}.]

Let (S_n) be the sum of the first (n) terms of the sequence ({1/2^n}). From the above, we see that (S_1=1/2), (S_2 = 3/4), etc. Our formula at the end shows that (S_n = 1-1/2^n).

Now consider the following limit:

[limlimits_{ntoinfty}S_n = limlimits_{ntoinfty}big(1-1/2^nbig) = 1.]

This limit can be interpreted as saying something amazing: the sum of all the terms of the sequence ({1/2^n}) is 1.} This example illustrates some interesting concepts that we explore in this section. We begin this exploration with some definitions.

Definition 31: Infinite Series, (n^text{th}) Partial Sums, Convergence, Divergence

Let ({a_n}) be a sequence.

1. The sum (sumlimits_{n=1}^infty a_n) is an infinite series (or, simply series).
2. Let ( S_n = sumlimits_{i=1}^n a_i); the sequence ({S_n}) is the sequence of (n^text{th}) partial sums of ({a_n}).
3. If the sequence ({S_n}) converges to (L), we say the series ( sumlimits_{n=1}^infty a_n) converges to (L), and we write ( sumlimits_{n=1}^infty a_n = L).
4. If the sequence ({S_n}) diverges, the series ( sumlimits_{n=1}^infty a_n) diverges.

Using our new terminology, we can state that the series ( sumlimits_{n=1}^infty 1/2^n) converges, and ( sumlimits_{n=1}^infty 1/2^n = 1.)

We will explore a variety of series in this section. We start with two series that diverge, showing how we might discern divergence.

Example (PageIndex{1}): Showing series diverge

1. Let ({a_n} = {n^2}). Show ( sumlimits_{n=1}^infty a_n) diverges.
2. Let ({b_n} = {(-1)^{n+1}}). Show ( sumlimits_{n=1}^infty b_n) diverges.

Solution

1. Consider (S_n), the (n^text{th}) partial sum. [begin{align*} S_n &= a_1+a_2+a_3+cdots+a_n &= 1^2+2^2+3^2cdots + n^2.end{align*}] By Theorem 37, this is [= frac{n(n+1)(2n+1)}{6}.] Since ( limlimits_{ntoinfty}S_n = infty), we conclude that the series ( sumlimits_{n=1}^infty n^2) diverges. It is instructive to write ( sumlimits_{n=1}^infty n^2=infty) for this tells us how the series diverges: it grows without bound.
   A scatter plot of the sequences ({a_n}) and ({S_n}) is given in Figure 8.7(a). The terms of ({a_n}) are growing, so the terms of the partial sums ({S_n}) are growing even faster, illustrating that the series diverges.
   
2. The sequence ({b_n}) starts with 1, (-1), 1, (-1), (ldots). Consider some of the partial sums (S_n) of ({b_n}): [begin{align*}S_1 &= 1S_2 &= 0S_3 &= 1S_4 &= 0end{align*}] This pattern repeats; we find that (S_n = left{begin{array}{cc} 1 & n text{ is odd} 0 & n text{ is even} end{array}right.)
   As ({S_n}) oscillates, repeating 1, 0, 1, 0, (ldots), we conclude that ( limlimits_{ntoinfty}S_n) does not exist, hence ( sumlimits_{n=1}^infty (-1)^{n+1}) diverges.
   A scatter plot of the sequence ({b_n}) and the partial sums ({S_n}) is given in Figure 8.7(b). When (n) is odd, (b_n = S_n) so the marks for (b_n) are drawn oversized to show they coincide.

While it is important to recognize when a series diverges, we are generally more interested in the series that converge. In this section we will demonstrate a few general techniques for determining convergence; later sections will delve deeper into this topic.

Geometric Series

One important type of series is a geometric series.

Definition 32: geometric series

A geometric series is a series of the form

[sumlimits_{n=0}^infty r^n = 1+r+r^2+r^3+cdots+r^n+cdots]

Note that the index starts at (n=0), not (n=1).

We started this section with a geometric series, although we dropped the first term of (1). One reason geometric series are important is that they have nice convergence properties.

theorem 60: convergence of geometric series

Consider the geometric series ( sumlimits_{n=0}^infty r^n).

1. The (n^text{th}) partial sum is: ( S_n = frac{1-r,^{n+1}}{1-r}).
2. The series converges if, and only if, (|r| < 1). When (|r|<1),
   [sumlimits_{n=0}^infty r^n = frac{1}{1-r}.]

According to Theorem 60, the series

[ sumlimits_{n=0}^infty frac{1}{2^n} =sumlimits_{n=0}^infty left(frac 12right)^2= 1+frac12+frac14+cdots]

converges as (r=1/2), and ( sumlimits_{n=0}^infty frac{1}{2^n} = frac{1}{1-1/2} = 2.) This concurs with our introductory example; while there we got a sum of 1, we skipped the first term of 1.

Example (PageIndex{2}): Exploring geometric series

Check the convergence of the following series. If the series converges, find its sum.

(1. sumlimits_{n=2}^infty left(frac34right)^n qquad 2. sumlimits_{n=0}^infty left(frac{-1}{2}right)^n qquad 3. sumlimits_{n=0}^infty 3^n)

Solution

1. Since (r=3/4<1), this series converges. By Theorem 60, we have that [sumlimits_{n=0}^infty left(frac34right)^n = frac{1}{1-3/4} = 4.] However, note the subscript of the summation in the given series: we are to start with (n=2). Therefore we subtract off the first two terms, giving:[sumlimits_{n=2}^infty left(frac34right)^n = 4 - 1 - frac34 = frac94.] This is illustrated in Figure 8.8.
   
2. Since (|r| = 1/2 < 1), this series converges, and by Theorem 60, [sumlimits_{n=0}^infty left(frac{-1}{2}right)^n = frac{1}{1-(-1/2)} = frac23.]
   The partial sums of this series are plotted in Figure 8.9(a). Note how the partial sums are not purely increasing as some of the terms of the sequence ({(-1/2)^n}) are negative.
   
3. Since (r>1), the series diverges. (This makes 'common sense'; we expect the sum [1+3+9+27 + 81+243+cdots] to diverge.) This is illustrated in Figure 8.9(b).

p-Series

Another important type of series is the p-series.

Definition 33: (p)-Series, General (P)-Series

1. A (p)--series is a series of the form [sumlimits_{n=1}^infty frac{1}{n^p}, qquad text{where (p>0).}]
2. A general (p)--series} is a series of the form
   [sumlimits_{n=1}^infty frac{1}{(an+b)^p}, qquad text{where (p>0) and (a), (b) are real numbers.}]

Like geometric series, one of the nice things about p--series is that they have easy to determine convergence properties.

theorem 61: convergence of general (P)--Series

A general (p)--series ( sumlimits_{n=1}^infty frac{1}{(an+b)^p}) will converge if, and only if, (p>1).

Note: Theorem 61 assumes that (an+bneq 0) for all (n). If (an+b=0) for some (n), then of course the series does not converge regardless of (p) as not all of the terms of the sequence are defined.

Example (PageIndex{3}): Determining convergence of series

Determine the convergence of the following series.

1. (sumlimits_{n=1}^infty frac{1}{n} )
2. (sumlimits_{n=1}^infty frac{1}{n^2})
3. (sumlimits_{n=1}^infty frac{1}{sqrt{n}})
4. (sumlimits_{n=1}^infty frac{(-1)^n}{n})
5. (sumlimits_{n=11}^infty frac{1}{(frac12n-5)^3})
6. (sumlimits_{n=1}^infty frac{1}{2^n} )

Solution

1. This is a (p)--series with (p=1). By Theorem 61, this series diverges.
   This series is a famous series, called the Harmonic Series, so named because of its relationship to harmonics in the study of music and sound.
2. This is a (p)--series with (p=2). By Theorem 61, it converges. Note that the theorem does not give a formula by which we can determine what the series converges to; we just know it converges. A famous, unexpected result is that this series converges to ( {pi^2}/{6}).
3. This is a (p)--series with (p=1/2); the theorem states that it diverges.
4. This is not a (p)--series; the definition does not allow for alternating signs. Therefore we cannot apply Theorem 61. (Another famous result states that this series, the Alternating Harmonic Series, converges to (ln 2).)
5. This is a general (p)--series with (p=3), therefore it converges.
6. This is not a (p)--series, but a geometric series with (r=1/2). It converges.

Later sections will provide tests by which we can determine whether or not a given series converges. This, in general, is much easier than determining what a given series converges to. There are many cases, though, where the sum can be determined.

Example (PageIndex{4}): Telescoping series

Evaluate the sum ( sumlimits_{n=1}^infty left(frac1n-frac1{n+1}right)).
Solution

It will help to write down some of the first few partial sums of this series.

[begin{align*}
S_1 &= frac11-frac12 & & = 1-frac12
S_2 &= left(frac11-frac12right) + left(frac12-frac13right) & & = 1-frac13
S_3 &= left(frac11-frac12right) + left(frac12-frac13right)+left(frac13-frac14right) & &= 1-frac14
S_4 &= left(frac11-frac12right) + left(frac12-frac13right)+left(frac13-frac14right) +left(frac14-frac15right)& &= 1-frac15
end{align*}]

Note how most of the terms in each partial sum are canceled out! In general, we see that ( S_n = 1-frac{1}{n+1}). The sequence ({S_n}) converges, as ( limlimits_{ntoinfty}S_n = limlimits_{ntoinfty}left(1-frac1{n+1}right) = 1), and so we conclude that ( sumlimits_{n=1}^infty left(frac1n-frac1{n+1}right) = 1). Partial sums of the series are plotted in Figure 8.10.

The series in Example 8.2.4 is an example of a telescoping series. Informally, a telescoping series is one in which the partial sums reduce to just a finite number of terms. The partial sum (S_n) did not contain (n) terms, but rather just two: 1 and (1/(n+1)).

When possible, seek a way to write an explicit formula for the (n^text{th}) partial sum (S_n). This makes evaluating the limit ( limlimits_{ntoinfty} S_n) much more approachable. We do so in the next example.

Note on notation: Most of the series we encounter will start with (n=1). For ease of notation, we will often write (sumlimits a_n) instead of writing ( sumlimits_{n=1}^infty a_n).

Example (PageIndex{5}): Evaluating series

Evaluate each of the following infinite series.

1. ( sumlimits_{n=1}^infty frac{2}{n^2+2n} qquad 2. sumlimits_{n=1}^infty lnleft(frac{n+1}{n}right))
Solution

1.8 Limits With Infinityap Calculus Pdf

1. We can decompose the fraction (2/(n^2+2n)) as [frac2{n^2+2n} = frac1n-frac1{n+2}.] (See Section 6.5, Partial Fraction Decomposition, to recall how this is done, if necessary.)
   Expressing the terms of ({S_n}) is now more instructive:
   [begin{align*}S_1 &= 1-frac13 &&= 1-frac13S_2 &= left(1-frac13right) + left(frac12-frac14right) &&= 1+frac12-frac13-frac14S_3 &= left(1-frac13right) + left(frac12-frac14right)+left(frac13-frac15right) &&= 1+frac12-frac14-frac15S_4 &= left(1-frac13right) + left(frac12-frac14right)+left(frac13-frac15right)+left(frac14-frac16right) &&= 1+frac12-frac15-frac16S_5 &= left(1-frac13right) + left(frac12-frac14right)+left(frac13-frac15right)+left(frac14-frac16right)+left(frac15-frac17right) &&= 1+frac12-frac16-frac17end{align*}]
   We again have a telescoping series. In each partial sum, most of the terms cancel and we obtain the formula ( S_n = 1+frac12-frac1{n+1}-frac1{n+2}.) Taking limits allows us to determine the convergence of the series:[limlimits_{ntoinfty}S_n = limlimits_{ntoinfty} left(1+frac12-frac1{n+1}-frac1{n+2}right) = frac32,quad text{so } sumlimits_{n=1}^infty frac1{n^2+2n} = frac32.]
   This is illustrated in Figure 8.11(a).
   
2. We begin by writing the first few partial sums of the series:
   [begin{align*}S_1 &= lnleft(2right) S_2 &= lnleft(2right)+lnleft(frac32right) S_3 &= lnleft(2right)+lnleft(frac32right)+lnleft(frac43right) S_4 &= lnleft(2right)+lnleft(frac32right)+lnleft(frac43right)+lnleft(frac54right) end{align*}]
   At first, this does not seem helpful, but recall the logarithmic identity: (ln x+ln y = ln (xy).) Applying this to (S_4) gives:[S_4 = lnleft(2right)+lnleft(frac32right)+lnleft(frac43right)+lnleft(frac54right) = lnleft(frac21cdotfrac32cdotfrac43cdotfrac54right) = lnleft(5right).]
   We can conclude that ({S_n} = big{ln (n+1)big}). This sequence does not converge, as ( limlimits_{ntoinfty}S_n=infty). Therefore ( sumlimits_{n=1}^infty lnleft(frac{n+1}{n}right)=infty); the series diverges. Note in Figure 8.11(b) how the sequence of partial sums grows slowly; after 100 terms, it is not yet over 5. Graphically we may be fooled into thinking the series converges, but our analysis above shows that it does not.

We are learning about a new mathematical object, the series. As done before, we apply 'old' mathematics to this new topic.

THEOREM 62 PROPERTIES OF INFINITE SERIES

Let (quad sumlimits_{n=1}^infty a_n = L,quad sumlimits_{n=1}^infty b_n = K), and let (c) be a constant.

1. Constant Multiple Rule: ( sumlimits_{n=1}^infty ccdot a_n = ccdotsumlimits_{n=1}^infty a_n = ccdot L.)
2. Sum/Difference Rule: ( sumlimits_{n=1}^infty big(a_npm b_nbig) = sumlimits_{n=1}^infty a_n pm sumlimits_{n=1}^infty b_n = L pm K.)

Before using this theorem, we provide a few 'famous' series.

KEY IDEA 31 IMPORTANT SERIES

1. ( sumlimits_{n=0}^infty frac1{n!} = e). (Note that the index starts with (n=0).)
2. ( sumlimits_{n=1}^infty frac1{n^2} = frac{pi^2}{6}).
3. ( sumlimits_{n=1}^infty frac{(-1)^{n+1}}{n^2} = frac{pi^2}{12}).
4. ( sumlimits_{n=0}^infty frac{(-1)^{n}}{2n+1} = frac{pi}{4}).
5. ( sumlimits_{n=1}^infty frac{1}{n} quad text{diverges}). (This is called the Harmonic Series.)
6. ( sumlimits_{n=1}^infty frac{(-1)^{n+1}}{n} = ln 2). (This is called the Alternating Harmonic Series.)

Example (PageIndex{6}): Evaluating series

1.8 Limits With Infinityap Calculus Solution

Evaluate the given series.

(1. sumlimits_{n=1}^infty frac{(-1)^{n+1}big(n^2-nbig)}{n^3}qquad 2. sumlimits_{n=1}^infty frac{1000}{n!}qquad 3. frac1{16}+frac1{25}+frac1{36}+frac1{49}+cdots)

Solution

1. We start by using algebra to break the series apart:
   [begin{align*}sumlimits_{n=1}^infty frac{(-1)^{n+1}big(n^2-nbig)}{n^3} &= sumlimits_{n=1}^inftyleft(frac{(-1)^{n+1}n^2}{n^3}-frac{(-1)^{n+1}n}{n^3}right) &= sumlimits_{n=1}^inftyfrac{(-1)^{n+1}}{n}-sumlimits_{n=1}^inftyfrac{(-1)^{n+1}}{n^2} &= ln(2) - frac{pi^2}{12} approx -0.1293.end{align*}]
   This is illustrated in Figure 8.12(a).
   
2. This looks very similar to the series that involves (e) in Key Idea 31. Note, however, that the series given in this example starts with (n=1) and not (n=0). The first term of the series in the Key Idea is (1/0! = 1), so we will subtract this from our result below:
   [begin{align*}sumlimits_{n=1}^infty frac{1000}{n!} &= 1000cdotsumlimits_{n=1}^infty frac{1}{n!} &= 1000cdot (e-1) approx 1718.28. end{align*}]
   This is illustrated in Figure 8.12(b). The graph shows how this particular series converges very rapidly.
   
3. The denominators in each term are perfect squares; we are adding ( sumlimits_{n=4}^infty frac{1}{n^2}) (note we start with (n=4), not (n=1)). This series will converge. Using the formula from Key Idea 31, we have the following:
   [begin{align*}sumlimits_{n=1}^infty frac1{n^2} &= sumlimits_{n=1}^3 frac1{n^2} +sumlimits_{n=4}^infty frac1{n^2} sumlimits_{n=1}^infty frac1{n^2} - sumlimits_{n=1}^3 frac1{n^2} &=sumlimits_{n=4}^infty frac1{n^2} frac{pi^2}{6} - left(frac11+frac14+frac19right) &= sumlimits_{n=4}^infty frac1{n^2} frac{pi^2}{6} - frac{49}{36} &= sumlimits_{n=4}^infty frac1{n^2} 0.2838&approx sumlimits_{n=4}^infty frac1{n^2} end{align*}]

It may take a while before one is comfortable with this statement, whose truth lies at the heart of the study of infinite series: it is possible that the sum of an infinite list of nonzero numbers is finite. We have seen this repeatedly in this section, yet it still may 'take some getting used to.'

As one contemplates the behavior of series, a few facts become clear.

1.8 Limits With Infinityap Calculus Answers

1. In order to add an infinite list of nonzero numbers and get a finite result, 'most' of those numbers must be 'very near' 0.
2. If a series diverges, it means that the sum of an infinite list of numbers is not finite (it may approach (pm infty) or it may oscillate), and:
   1. The series will still diverge if the first term is removed.
   2. The series will still diverge if the first 10 terms are removed.
   3. The series will still diverge if the first (1,000,000) terms are removed.
   4. The series will still diverge if any finite number of terms from anywhere in the series are removed.

These concepts are very important and lie at the heart of the next two theorems.

theorem 63 (n^text{th})--Term Test for Convergence/Divergence

1.8 Limits With Infinityap Calculus Solutions

Consider the series ( sumlimits_{n=1}^infty a_n).

1. If ( sumlimits_{n=1}^infty a_n) converges, then ( limlimits_{ntoinfty}a_n =0).
2. If ( limlimits_{ntoinfty}a_n neq 0), then ( sumlimits_{n=1}^infty a_n) diverges.

Note that the two statements in Theorem 63 are really the same. In order to converge, the limit of the terms of the sequence must approach 0; if they do not, the series will not converge.

Looking back, we can apply this theorem to the series in Example 8.2.1. In that example, the (n^text{th}) terms of both sequences do not converge to 0, therefore we can quickly conclude that each series diverges.

Important! This theorem does not state that if ( limlimits_{ntoinfty} a_n = 0) then ( sumlimits_{n=1}^infty a_n ) converges. The standard example of this is the Harmonic Series, as given in Key Idea 31. The Harmonic Sequence, ({1/n}), converges to 0; the Harmonic Series, ( sumlimits_{n=1}^infty 1/n), diverges.

theorem 64 infinite nature of series

The convergence or divergence remains unchanged by the addition or subtraction of any finite number of terms. That is:

1. A divergent series will remain divergent with the addition or subtraction of any finite number of terms.
2. A convergent series will remain convergent with the addition or subtraction of any finite number of terms. (Of course, the sum will likely change.)

Consider once more the Harmonic Series ( sumlimits_{n=1}^infty frac1n) which diverges; that is, the sequence of partial sums ({S_n}) grows (very, very slowly) without bound. One might think that by removing the 'large' terms of the sequence that perhaps the series will converge. This is simply not the case. For instance, the sum of the first 10 million terms of the Harmonic Series is about 16.7. Removing the first 10 million terms from the Harmonic Series changes the (n^text{th}) partial sums, effectively subtracting 16.7 from the sum. However, a sequence that is growing without bound will still grow without bound when 16.7 is subtracted from it.

The equations below illustrate this. The first line shows the infinite sum of the Harmonic Series split into the sum of the first 10 million terms plus the sum of 'everything else.' The next equation shows us subtracting these first 10 million terms from both sides. The final equation employs a bit of 'psuedo--math': subtracting 16.7 from 'infinity' still leaves one with 'infinity.'

[begin{align*}sumlimits_{n=1}^infty frac1n &= sumlimits_{n=1}^{10,000,000}frac1n quad + sumlimits_{n=10,000,001}^infty frac1n sumlimits_{n=1}^infty frac1n - sumlimits_{n=1}^{10,000,000}frac1n &= sumlimits_{n=10,000,001}^infty frac1n infty - 16.7 &= infty.end{align*} ]

This section introduced us to series and defined a few special types of series whose convergence properties are well known: we know when a (p)-series or a geometric series converges or diverges. Most series that we encounter are not one of these types, but we are still interested in knowing whether or not they converge. The next three sections introduce tests that help us determine whether or not a given series converges.

Contributors and Attributions

• Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/