Convergence of Infinite Series

What exactly does the convergence of an infinite series tell us? What does it mean? The goal of this tutorial is to guide you through the processes used to analyze the convergence of an infinite series, and to uncover its importance in relation to other areas of mathematices.

This begins our study of whether a given series actually has a value - that is, whether its terms get small fast enough that infinitely many terms can be added to get a finite answer. In a nutshell, this explains the definitions of the two terms: convergence and divergence. Does it seem strange to you that you can add up an infinite number of positive numbers and get a finite answer? Well, if it does, you're not alone. This confuses a lot of students when first faced with this concept. A first impression might be that the result of the sum must get bigger and bigger all the time. However, what if we keep adding a smaller and smaller term each time? Will this give us a finite answer? Well, these are competing processes, and this tutorial will try to introduce you to the tools which will enable you to decide whether the infinite sum gives a finite answer or whether it grows without bound.

First, consider the following sum:

Textbooks would introduce the compact summation notation . Now, common sense would tell us that the infinite sum would have a 1 in every decimal place, giving , which you may recall is . So, this simple example shows that an infinite series with all positive terms can have a finite answer.

With this little experience, let us consider the next two examples to see how careful we need to be. We will show (graphically) in these two examples, that just because the terms in an infinite series get smaller, it does not mean the series will converge. So, to make this as painless as possible, let's picture the graph of these two similiar infinite series and see what happens. The two series we will look at are and . Click HERE to see the two series graphed on the same axis. Comparing the two infinite series will help us picture the possible convergence of the two different series.

Notice on this graph, you can tell right away that is leveling off to a specific value (). This is a strong indication that the series is going to converge. Now, the other series, , does not seem to converge at all. It's values are getting larger and larger with no leveling off to a specific value.

In order to be sure, we need to take more terms of the series and see what happens. Click HERE to see more terms of the series. On this graph, we have taken 100 terms in the series. As we can see there is even a stronger indication that does indeed converge. Now let's look at the other series, . See how taking more terms of the series causes the sum to get larger and larger without bound? This is an indication that the series diverges. It will never reach a specific value and level off as the convergent series did. The series is called the harmonic series and it can be proven to be divergent. For a proof of the divergence of click HERE . For a proof of the convergence of , click HERE . This shows that even though two infinite series may be similiar, they can behave in completely opposite ways. This also proves our point that just because the terms in a series get smaller all the time, this doesn't necessarily mean that the series will converge to a finite number.

There are various tests that can be applied to an infinite series to determine whether or not it converges. The first, and most often used, is called the Ratio Test. The Ratio Test can be thought of as a measurement of the rate of growth (or decline) and this can be found by examining the ratio as .

Let be an infinite series and suppose that

Then
a) the series converges if ,
b) the series diverges if ,
c) the series may converge or it may diverge if . The test will provide no information for this case.

Okay, let's stop here a second and think about the definition in a less formal way. Since convergence is affected by the rate of shrinkage of the terms in the series, considering the ratio will be a big help in analyzing our series. Say we have an infinite series, such as:

Simple, right? This is just the series . Now, the first thing we want to know is whether or not the terms in the series are getting small for large values of n. Let's look at the partial sum where p = 2. Then our two terms are 1 and . If the series converged, then we would expect to be less than 1, right? This does not turn out to be the case here since . This is because only the tail of the series counts in our analysis of how the series will behave. As long as the series eventually converges, it doesn't matter whether the terms in the series at the beginning are getting large or small.

Now, how can we determine how the tail of the series is behaving? We can do so by taking the ratio of the term and the term. Recall that it is important for the terms in the series to be getting small fast enough for the sum to converge. What do we do about this? Well, by taking the limit as of our ratio, we can see the rate of convergence. Let's go ahead and test for convergence of our series. So, our term for this series is while our term is . Then the ratio test tells us:

Cancelling appropriate terms leaves us with:

Next, divide all the terms in the ratio by the highest power of n and take the limit:

Therefore, by the ratio test, the series does in fact converge.

IMPORTANT HINT: The ratio test is especially useful for infinite series with exponents, such as the example above. It is also particularly convenient when the infinite series contains a factorial. The ratio test makes it possible for the exponents and the factorials to be cancelled, leaving terms that are easier to deal with.

Now, let's go back and look at our two earlier series for comparison: and .

Using the ratio test, determine whether or not the series converges. So, we must take the where and are the absolute values of the terms in the series. So, we get:

Dividing all the terms by the highest power of n , (in our case just n ), leaves us with:

Therefore, we are in the case where the ratio test gives no information (). We must use a different test to determine convergence. Okay, let's see if we can use the ratio test to determine the convergence of our other series, . So,

By expanding , we are left with the limit:

By dividing by the highest power of n, (in our case ), we have:

Once again, the ratio test gives no information for the convergence of this series.

For clarification purposes, the formal definition of the convergence of an infinite series can be explained using the concept of limits. Consider the partial sum:

Now, if we let , we get the infinite series in the general form:

This series has no hope of being convergent unless the sequence has . That is, take the limit of the terms: . In order for to converge, as a minimum requirement, we must have as . There is no hope of getting a finite sum from infinitely many terms unless those terms get small. If the terms do not approach zero, the series cannot have a finite sum. Therefore the series must diverge . Divergence means that the terms of the series will grow without bound as .

Let's take a breather for a moment and try solving a couple of infinite series problems using the Ratio Test.

1.) Use the Ratio Test to investigate the convergence of the following series:

Click HERE for the result.

2.) Use the Ratio Test to investigate the convergence another series:

Click HERE for the result.

What if we had a power series? Would the ratio test still be useful in determining whether or not a given power series will converge? Okay, before we get ahead of ourselves, we must first understand what a power series is. A power series is simply one type of infinite series that has the form:

where the coefficients are constants. This is called a power series in x (1). A more general form of a power series is:

where a is called the center. Such a series is called a power series in (x - a) (2). Note: this form of the equation is the same as the first if we take a = 0. Or, if you expand out the terms of the second equation and create new constant coefficients, we also get the first equation. However, sometimes it is more convenient to write a power series in the form of the second equation.

This is now a good opportunity to discuss whether a power series will converge for all values of x. Or, if a given power series does not converge for all values of x, can we find the values where it does converge? The values where the power series converge are described by using the two terms, radius of convergence and the interval of convergence. Using the first form of the power series equation, the radius of convergence is a non-negative number R which has the property that if , then the series converges for this x and if , then the series diverges for this x. If , the series may converge or diverge. Then the interval of convergence is defined as the interval . The brackets can be open or closed depending on whether or not the series converges or diverges for the two endpoints, -R and R (2). We will get into methods for determining this later. NOTE: If we used the second form of the equation, just replace x by x - a and solve to find the radius of convergence. So, our radius of convergence is defined by .

Now comes the time where we solidify the concept of radius of convergence and interval of convergence of a power series by determining the methods used to find them. What if we had a power series of the form:

Now let's take x = 5. Will the series converge? Let's use the Ratio Test in order to find out. So, our series becomes:

Taking the limit of the ratio of the terms gives us:

Now, divide the numerator and denominator of our ratio by the highest power of n:

Therefore, this series does not seem to converge. In fact, this example shows that it diverges. What if we take another value for x, say ? Now we have the series:

and the ratio test tells us:

Again, divide both numerator and denominator by the highest power of n:

Once again the ratio test is telling us that this series diverges. Let's try one last value of x, say . Then our new series is:

and the ratio test tells us:

Divide both the numerator and denominator by the highest power of n:

Wait a minute, what happened here? All of a sudden our series is converging! Why are we getting the series converging for different values of x? This suggests that it is possible for a power series to converge for certain values of x and diverge for other values. It turns out that this is precisely the case. This is an amazing characteristic about a power series. Now the question that remains is just this: Is there a way to determine the values where a given power series will converge? If your instincts tell you that there is, then they are exactly correct! Okay, go ahead and brainstorm for a minute about possible ways to find these values.

Did you come up with any? Well, instead of picking values for x , why don't we leave our x value in our ratio when we use the test? Then when we have simplified the ratio as much as possible, we will see if we can find the largest value of x which makes the ratio less than 1. If we can, then for all smaller values of x, the series will converge. This will give us our interval of convergence. Does this make any sense? Let's clarify this by using our above power series as an example. So, our power series is:

and the ratio test gives:

Now, divide the numerator and denominator of our ratio by the highest power of n:

Therefore, our power series should converge for the values where . The radius of convergence tells us the range of values where the power series will converge. In this case, our radius of convergence would be R = 1. The interval of convergence will tell us the interval where the power series will converge. In our case, it would be or . Recall that the brackets for our interval of convergence may be open or closed depending on whether the series will converge or diverge at the two endpoints 1 and -1. How do we determine this? Well, all we need to do is plug in our two values for x in our power series and determine the state of its convergence. Let's start off by using the ratio test. So, for x = 1, our power series takes the form:

and the ratio test gives:

This tells us that we will not be able to use the ratio test when determining the state of convergence of the endpoints. In fact, it turns out that this is the case with all power series. For this simple case, we notice that so the series must diverge. However, it is useful to learn other tests of convergence for the endpoints of a power series and for other series that cannot be determined by the ratio test.

Before, we go into other tests of convergence for an infinite series, let's practice some problems using the ratio test on a power series.

1.) Find the open interval of convergence of the series:

Click HERE for the solution.

2.) Find the open interval of convergence of the series:

Click HERE for the solution.

3.) Find the open interval of convergence of the series:

Click HERE for the solution.

Okay, let's learn some more tests for the convergence of an infinite series to apply to the endpoints of an interval of convergence and when we are not able to get a solution using the Ratio Test (e.g. in the case). The test I would like to present first is called the - Term Test for Divergence . It should be one of the first tests you should apply when faced with a problem. Its simplicity makes it possible to analyze certain types of problems almost instantly.

The - Term Test for Divergence:

** If , or if fails to exist, then diverges. **

or stated another way:

** If converges, then . **

In other words, in order for a series to be convergent, the limit of your terms need to go to 0 as . There are three main cases for a divergent series: (1) the series could go to , (2) the sequence goes to a constant value, (3) the limit of the term does not exist (2). Let's follow through an example of each case to see what I mean.

Case (1):

Well, in this case the . Therefore, the series diverges.

Case (2):

Divide the numerator and denominator by the highest power of n and then take the limit:

Therefore, the series diverges by the term test. Let's look at this another way. Notice how every term in our series is greater than ? No matter what value you take for n when , you always get a value that is greater than . Adding an infinite number of terms will be greater than . Clearly this will diverge. Well, any series that contains a sequence of numbers greater than a series that diverges will diverge as well.

Case (3):

This series diverges because the limit does not exist.

NOTE: If you use the term test and your limit turns out to be 0, you must use another test to determine convergence.

The second test I want to talk about is called the Integral Test. This test becomes handy when you don't know the behavior of the actual series, but you do know the properties of the corresponding integral. Let's state the Integral Test and then go through an example to see what I mean.

NOTE: As you may recall, the Integral Test was already introduced when we proved the convergence of the series .

Integral Test:

** Let where is a continuous, positive, decreasing function of x for all . Then the series and the integral both converge or both diverge (2). **

Fascinating, eh? So, if you know how an integral behaves, then you automatically know how its corresponding series defined on integer values will behave. We saw an example of this when we proved the convergence of the series . Let's look at another example for clarification purposes. We could have proved the divergence of in a manner similiar to what we did with by using the integral test. Here, instead of restricting n to integer values, make it correspond to a continuous variable x. Then the function will give the same values as when x is an integer. Before we look at the integral test let's look at a graph of the Riemann Sums of the corresponding integral. Click HERE to see the graph of the Riemann Sums. We know that the integral will be greater than the green Riemann sums (which are the same as the series), so if the integral converges the series converges as well. On the other hand, the integral is less than the dotted blue Riemann sums (which are also the same as the series, just rearranged), so if the integral diverges, the series diverges as well. Let's work this problem out. So, the corresponding improper integral is:

and evaluate. So:

Well, evaluating the integral leaves us with:

Therefore,

and the integral diverges. This means by the integral test that the corresponding series diverges as well.

Limit Comparison Test:

** a) Test for convergence. If for and there is a convergent series such that and

then converges. **

** b) Test for divergence. If for and there is a divergent series such that and

then diverges (2). **

Once again, let's work through an example in an attempt at understanding this test.

In determining convergence or divergence, only the tails count. When n is very large, the highest powers of n in numerator and denominator are what count the most. So, from our example, set

and notice that it behaves approximately like = . So, by comparing it to , we can make the educated guess that our example will diverge (since diverges). So, we take

and

and look at the ratio

As , the . Therefore,

diverges (2).

EXTRA MIXED PROBLEMS

In this section, this tutorial will provide you with some mixed practice problems for you to try. The solutions will be provided for you but try and do them on your own first.

Problem 1.)

Determine whether or not the following infinite series will converge or diverge:

For the solution Click HERE .

Problem 2.)

Determine whether or not the following infinite series will converge or diverge:

For the solution Click HERE .

Problem 3.)

Find the radius and interval of convergence of the power series:

Then find out whether or not the endpoints of our interval of convergence will diverge or converge. Remember: All you need to do to figure this out is plug in your endpoints for x. This will create a new infinite series. Next, using a method other than the ratio test, determine convergence or divergence.

For the solution to this problem Click HERE .

Problem 4.)

Determine whether or not the following infinite series converges:

Click HERE for the solution.

This concludes this tutorial. If you would like to see another tutorial where one of the topics is determining the radius of convergence and interval of convergence is determined on a Taylor Series approximation Click HERE for my Taylor Series tutorial.

NOTES:

This page was written by Jennifer J. Pearce in Summer of 1996 under the guidance of Prof. Kevin Short at the University of New Hampshire

Jennifer J Pearce
Wed Oct 30 20:39:43 EST 1996