# 7 Series

A series is written as

and the definitions of convergent and divergent series are stated by Larson and Edwards as follows [15].

Definition I.14
For the infinite series , the nth partial sum is

If the sequence of partial sums converges to , then the series converges. The limit is called the sum of the series.

If diverges, then the series diverges.

As with sequences, we are concerned with whether a series converges or diverges. There are some types of series that have their own set of rules for whether they converge or not, such as geometric series and p-series. If a series is not one of these special types, then we have tests that can be used to determine convergence, such as the Direct Comparison and Alternating Series tests.

#### Geometric Series

A geometric series is a special type of series. Not only is it easy to tell whether such a series converges or diverges, but it is also easy to find the sum that the series converges to. Let’s look at the following theorem and proof from Larson and Edwards [15].

Theorem I.15 (Convergence of a Geometric Series)

A geometric series with ratio diverges when . If , then the series converges to the sum

Proof.
It is easy to see that the series diverges when If , then we have the partial sum

When , it follows that . So,

Thus, the series converges, and its sum is .

When , it follows that alternates between and , and when , it follows that . So if

Thus, the limit does not exist, and the series diverges.

Let’s now do an example I completed for Calculus II that comes from Larson and Edwards [15].

Example 15
Does the series converge or diverge? If it converges, what is the sum?Solution
We have that and . Because
the series converges, and the sum is

by Theorem I.15.

Another special type of series for which it is easy to determine convergence and divergence is the -series.

#### -series

Consider the following theorem provided by Larson and Edwards [15].

Theorem I.16

The -series

converges for and diverges for .

Let’s now look at how to use this theorem. This problem is one I completed for Calculus II and comes from Larson and Edwards [15].

Example 16

Does converge or diverge?

Solution

Since and , this series converges by Theorem I.16.

Because many series do not fall into nice categories such as geometric series and -series, we need tests that we can use to help us determine convergence and divergence.

#### Direct Comparison Test

The first test for convergence and divergence that we are going to discuss is the direct comparison test. This test is stated as a theorem by Larson and Edwards as follows [15].

Theorem I.17

Let for all .

1. If converges, then converges.
2. If diverges, then diverges.

Proof.
To prove the first part, let and let

and

Since , converges to by Definition I.14.  Additionally, because for all , we have

and

So, the sequence is bounded above. Because , is also increasing and thus bounded below by . By Theorem I.13, converges, and converges by Definition I.14.

Therefore, if for all and converges, converges as well. Because part 2 is the contrapositive statement of part 1, part 2 is also true. (See Part II: Chapter 11.)

This proof is one I completed as a challenge for myself with some help from the proof given for this test in Larson and Edwards [15]. Let’s now work through an example I completed for Calculus II that comes from Larson and Edwards [15].

Example 17

Using the Direct Comparison Test, does converge or diverge?

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Solution
The series

looks very similar to the geometric series

We know that this geometric series converges by Theorem I.15 so the Direct Comparison test will help us only if for all ,

We can show this is true the following way.

Because for all and converges, converges by the Direct Comparison Test.

We are now going to discuss one other test, the alternating series test, before concluding our discussion of series.

#### Alternating Series Test

The alternating series test helps us determine if a series is converging. Larson and Edwards state the test as follows [15].

Theorem I.18 (Alternating Series Test)

Let . The alternating series

converge when the two conditions listed below are met.

1. , for all

Proof.
Consider the alternating series . For this series, the partial sum (where is even)

has all nonpositive terms because , and therefore is a nonincreasing sequence. But the partial sum can be rewritten as

which implies that for every integer (because ). So, is a bounded, nonincreasing sequence that converges to some value by Theorem I.13. Notice that . Because , So we have,

Because both and converge to the same limit , it follows that also converges to . Consequently, the given alternating series converges by Definition I.14.

This proof is one I completed for Calculus II. The proof for when the series is

can be done in a similar manner and is found in Larson and Edwards [15]. Let’s now work through an example I completed for Calculus II that comes from Larson and Edwards [15].

Example 18

Does converge or diverge?

Solution
Because

and , we need to check the two conditions of the Alternating Series test to see if this test applies. Because

the first condition holds. Additionally, because

for all , the second condition holds. Therefore, converges by the Alternating Series Test.

Our final calculus topic we will discuss is differential equations. Differential equations contain derivatives within them, and in order to solve them, we must use integration. There is so much we could discuss, but we are going to settle with one specific type, the separable differential equation.