Given the sequence $\{a_n\} = \{1/2^n\} = 1/2,\ 1/4,\ 1/8,\ \ldots\text{,}$ consider the following sums: \begin{equation*} \begin{array}{ccccc} a_1 \amp =\amp 1/2 \amp =\amp 1/2\\ a_1+a_2 \amp =\amp 1/2+1/4 \amp =\amp 3/4\\ a_1+a_2+a_3 \amp =\amp 1/2+1/4+1/8 \amp =\amp 7/8\\ a_1+a_2+a_3+a_4 \amp =\amp 1/2+1/4+1/8+1/16 \amp =\amp 15/16 \end{array} \end{equation*}

In general, we can show that \begin{equation*} a_1+a_2+a_3+\cdots +a_n = \frac{2^n-1}{2^n} = 1-\frac{1}{2^n}. \end{equation*}

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

Now consider the following limit: $\lim\limits_{n\to\infty}S_n = \lim_{n\to\infty}\big(1-1/2^n\big) = 1\text{.}$ 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.

##### Definition8.2.1Infinite Series, $n^th$ Partial Sums, Convergence, Divergence

Let $\{a_n\}$ be a sequence.

1. The sum $\ds \infser a_n$ is an infinite series (or, simply series).

2. Let $\ds S_n = \sum_{i=1}^n a_i$ ; the sequence $\{S_n\}$ is the sequence of $n^\text{ th }$ partial sums of $\{a_n\}\text{.}$

3. If the sequence $\{S_n\}$ converges to $L\text{,}$ we say the series $\ds \infser a_n$ converges to $L\text{,}$ and we write $\ds \infser a_n = L\text{.}$

4. If the sequence $\{S_n\}$ diverges, the series $\ds \infser a_n$ diverges.

Using our new terminology, we can state that the series $\ds \infser 1/2^n$ converges, and $\ds \infser 1/2^n = 1\text{.}$

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

##### Example8.2.2Showing series diverge
1. Let $\{a_n\} = \{n^2\}\text{.}$ Show $\ds \infser a_n$ diverges.

2. Let $\{b_n\} = \{(-1)^{n+1}\}\text{.}$ Show $\ds \infser b_n$ diverges.

Solution

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.

# Subsection8.2.1Geometric Series

One important type of series is a geometric series.

##### Definition8.2.4Geometric Series

A geometric series is a series of the form \begin{equation*} \infser[0] r^n = 1+r+r^2+r^3+\cdots+r^n+\cdots \end{equation*}

Note that the index starts at $n=0\text{,}$ not $n=1\text{.}$

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

According to Theorem 8.2.5, the series \begin{equation*} \ds\infser[0] \frac{1}{2^n} =\infser[0] \left(\frac 12\right)^2= 1+\frac12+\frac14+\cdots \end{equation*} converges as $r=1/2 \lt 1\text{,}$ and $\ds \infser[0] \frac{1}{2^n} = \frac{1}{1-1/2} = 2\text{.}$ This concurs with our introductory example; while there we got a sum of 1, we skipped the first term of 1.

##### Example8.2.6Exploring geometric series

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

1. $\ds \sum_{n=2}^\infty \left(\frac34\right)^n$

2. $\ds \infser[0] \left(\frac{-1}{2}\right)^n$

3. $\ds \infser[0] 3^n$

Solution

# Subsection8.2.2$p$–Series

Another important type of series is the p-series.

##### Definition8.2.9$p$–Series, General $p$–Series
1. A $p$–series is a series of the form \begin{equation*} \infser \frac{1}{n^p}, \qquad \text{ where $p>0$. } \end{equation*}

2. A general $p$–series is a series of the form \begin{equation*} \infser \frac{1}{(an+b)^p}, \qquad \text{ where $p>0$ and $a$, $b$ are real numbers. } \end{equation*}

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

We will be able to prove Theorem 8.2.10 in Section 8.3. This theorem assumes that $an+b\neq 0$ for all $n\text{.}$ If $an+b=0$ for some $n\text{,}$ then of course the series does not converge regardless of $p$ as not all of the terms of the sequence are defined.

##### Example8.2.11Determining convergence of series

Determine the convergence of the following series.

1. $\ds\infser \frac{1}{n}$

2. $\ds\infser \frac{1}{n^2}$

3. $\ds\infser \frac{1}{\sqrt{n}}$

4. $\ds\infser \frac{(-1)^n}{n}$

5. $\ds\sum_{n=11}^\infty \frac{1}{(\frac12n-5)^3}$

6. $\ds\infser \frac{1}{2^n}$

Solution

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.

##### Example8.2.12Telescoping series

Evaluate the sum $\ds \infser \left(\frac1n-\frac1{n+1}\right)\text{.}$

Solution

The series in Example 8.2.12 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 $1/(n+1)\text{.}$

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

##### Example8.2.14Evaluating series

Evaluate each of the following infinite series.

1. $\ds \infser \frac{2}{n^2+2n}$

2. $\ds \infser \ln\left(\frac{n+1}{n}\right)$

Solution

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

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

##### Key Idea8.2.17Important Series
1. $\ds\infser[0] \frac1{n!} = e\text{.}$ (Note that the index starts with $n=0\text{.}$)

2. $\ds\infser \frac1{n^2} = \frac{\pi^2}{6}\text{.}$

3. $\ds\infser \frac{(-1)^{n+1}}{n^2} = \frac{\pi^2}{12}\text{.}$

4. $\ds\infser[0] \frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}\text{.}$

5. $\ds\infser \frac{1}{n}$ diverges. (This is called the Harmonic Series.)

6. $\ds\infser \frac{(-1)^{n+1}}{n} = \ln(2)\text{.}$ (This is called the Alternating Harmonic Series.)

##### Example8.2.18Evaluating series

Evaluate the given series.

1. $\ds\infser \frac{(-1)^{n+1}\big(n^2-n\big)}{n^3}$

2. $\ds\infser \frac{1000}{n!}$

3. $\ds \frac1{16}+\frac1{25}+\frac1{36}+\frac1{49}+\cdots$

Solution

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. 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.

Note that the two statements in Theorem 8.2.20 are logically equivalent (each is the other's contrapositive). 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 8.2.2. 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 $\lim\limits_{n\to\infty} a_n = 0$ then $\ds \infser a_n$ converges. The standard example of this is the Harmonic Series, as given in Key Idea 8.2.17. The Harmonic Sequence, $\{1/n\}\text{,}$ converges to 0; the Harmonic Series, $\ds \infser 1/n\text{,}$ diverges.

Consider once more the Harmonic Series $\ds\infser \frac1n$ 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*} \infser \frac1n \amp = \sum_{n=1}^{10,000,000}\frac1n + \ds\sum_{n=10,000,001}^\infty \frac1n\\ \infser \frac1n - \sum_{n=1}^{10,000,000}\frac1n\amp = \ds\sum_{n=10,000,001}^\infty \frac1n\\ \infty - 16.7 \amp = \infty. \end{align*}

Just for fun, we can show that the Harmonic Series diverges algebraically (without the use of Convergence of General $p$–Series).

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.

# Subsection8.2.3Exercises

Terms and Concepts

In the following exercises, a series $\ds\infser a_n$ is given.