8.2: Infinite Series
Learning Objectives
 Demonstrate an understanding of series and the sequence of partial sums
 Recognize geometric series and determine when they converge or diverge
 Compute the sum of a convergent geometric series
 Determine convergence or divergence of series using the nthTerm Test
Infinite Series (series, sequence of partial sums, convergence, divergence)
Series
Another topic that involves an infinite number of terms is the topic of infinite series. We can represent certain functions and numbers with an infinite series. For example, any real number that can be written as a nonterminating decimal can be represented as an infinite series.
Example 1
The rational number
On the other hand, the number
Do you see the difference between an infinite series and a finite series? Let’s define what we mean by an infinite series.
Infinite Series
An infinite series is the sum of an infinite number of terms,
A shorthand notation for an infinite series is to use sigma notation:
We can make finite sums from the terms of the infinite series:
The first sum is the first term of the sequence. The second sum is the sum of the first two terms. The third term is the sum of the first three terms. Thus, the
Sequence of Partial Sums
As you can see, the sums
Partial Sums
For an infinite series
The sequence
Example 2
Find the first five partial sums of the infinite series
Solution
To further explore series, try experimenting with this applet. The applet shows the terms of a series as well as selected partial sums of the series. Series Applet. As you see from this applet, for some series the partial sums appear to approach a fixed number, while for other series the partial sums do not. Exploring this phenomenon is the topic of the next sections.
Convergence and Divergence
Just as with sequences, we can talk about convergence and divergence of infinite series. It turns out that the convergence or divergence of an infinite series depends on the convergence or divergence of the sequence of partial sums.
Convergence/Divergence of Series
Let
If
Example 3
Does the infinite series
Solution
To make our work easier, write the infinite series
To solve for convergence or divergence of the infinite series, write the formula for the
It is rather difficult to find
First, multiply both sides of the equation
Now we have two equations:
Subtract the bottom equation from the top equation to cancel terms and simplifying:
Solve for
Now we find the limit of both sides:
The sum of the infinite series is
Geometric Series
The geometric series is a special kind of infinite series whose convergence or divergence is based on a certain number associated with the series.
Geometric Series
A geometric series is an infinite series written as
In sigma notation, a geometric series is written as
The number
Example 4
Here are some examples of geometric series.
Geometric Series 












The convergence or divergence of a geometric series depends on
Theorem
Suppose that the geometric series
 The geometric series converges if
r<1 and the sum of the series isa1−r .  The geometric series diverges if
r≥1 .
Example 5
Determine if the series
Solution
The series is a geometric series that can be written as
Example 6
Determine if the series
Solution
The series is a geometric series with
Example 7
Determine if
Solution
If we rewrite the series in terms of powers of
It looks like a geometric series with
However, if we write the definition of a geometric series for
The original problem,
The sum of the series is:
Other Convergent Series
There are other infinite series that will converge.
Example 8
Determine if
Solution
The
We can simplify
will cancel out. Continue in this way to cancel opposite terms. This sum is a telescoping sum, which is a sum of terms that cancel each other out so that the sum will fold neatly like a folding telescope. Thus, we can write the partial sum as
Then
Other Divergent Series (
Determining convergence by using the limit of the sequence of partial sums is not always feasible or practical. For other series, it is more useful to apply tests to determine if an infinite series converges or diverges. Here are two theorems that help us determine convergence or divergence.
Theorem (The
If the infinite series
Theorem
If
The first theorem tells us that if an infinite series converges, then the limit of the sequence of terms is
The second theorem tells us that if limit of the sequence of terms is not zero, then the infinites series diverges. This gives us the first test of divergence: the
Example 9
Determine if
Solution
We can use the
Because
Example 10
Determine if \begin{align*}\sum_{k=1}^\infty \frac{8}{k3}\end{align*}
Solution
Using the \begin{align*}n\end{align*}
Rules for Convergent Series, Reindexing
Rules
As with sequences, there are rules for convergent infinite series that help make it easier to determine convergence.
Theorem (Rules for Convergent Series)
1. Suppose \begin{align*}\sum_{k=1}^\infty u_k\end{align*}
Then \begin{align*}\sum_{k=1}^\infty (u_k+v_k)\end{align*}
\begin{align*}\sum_{k=1}^\infty (u_k+v_k) =\sum_{k=1}^\infty (u_k)+\sum_{k=1}^\infty (v_k)=S_1+S_2\end{align*}
(The sum or difference of convergent series is also convergent.)
2. Let \begin{align*}c \neq 0\end{align*}
Suppose \begin{align*}\sum_{k=1}^\infty u_k\end{align*}
\begin{align*}\sum_{k=1}^\infty cu_i = c\sum_{k=1}^\infty u_k = cS\end{align*}
If \begin{align*}\sum_{k=1}^\infty u_k\end{align*}
(Multiplying by a nonzero constant does not affect convergence or divergence.)
Example 10
Find the sum of \begin{align*}\sum_{k=1}^\infty \left(\frac{2}{3^{k1}}+\frac{1}{8^{k1}}\right)\end{align*}
Solution
Using the Rules Theorem, \begin{align*}\sum_{k=1}^\infty \left(\frac{2}{3^{k1}}+\frac{1}{8^{k1}}\right) = \sum_{k=1}^\infty \frac{2}{3^{k1}}+ \sum_{k=1}^\infty \frac{1}{8^{k1}}\end{align*}
\begin{align*}\sum_{k=1}^\infty \frac{2}{3^{k1}}\end{align*}
\begin{align*}\sum_{k=1}^\infty \frac{1}{8^{k1}}\end{align*}
Then \begin{align*}\sum_{k=1}^\infty \left(\frac{2}{3^{k1}} + \frac{1}{8^{k1}}\right) = \sum_{k=1}^\infty \frac{2}{3^{k1}}+ \sum_{i=1}^\infty \frac{1}{8^{k1}} = 3 + \frac{8}{7} = \frac{29}{7}\end{align*}
Example 11
Find the sum of \begin{align*}\sum_{k=1}^\infty 2\left(\frac{5}{6^{k1}}\right)\end{align*}
Solution
By the rules for constant in infinite series, \begin{align*}\sum_{k=1}^\infty 2\left(\frac{5}{6^{k1}}\right) = 2 \sum_{k=1}^\infty \frac{5}{6^{k1}}\end{align*}
Then \begin{align*}\sum_{k=1}^\infty 2\left(\frac{5}{6^{k1}}\right) = 2 \times 6 = 12\end{align*}
Adding or subtracting a finite number of terms from an infinite series does not affect convergence or divergence.
Theorem
If \begin{align*}\sum_{k=1}^\infty u_k\end{align*}
If \begin{align*}\sum_{k=1}^\infty u_k\end{align*}
Likewise, if \begin{align*}\sum_{k=1}^\infty u_k\end{align*}