Series

Calculus 2

From Sequences to Series

A sequence is a list of numbers written in a specific order. A series is simply the summation of the terms of a sequence. If the sequence is finite, we end up with a finite series. If the sequence is infinite, we end up with an infinite series.

We define the partial sum $S_n$ as the sum of the first $n$ terms in a sequence. For example, given the sequence $a_n = \{2,5,8,11, \dots\}$ , we have the following partial sums:

\begin{aligned} S_1 &= a_1 = 2 \\ S_2 &= a_1 + a_2 = 2+5=7 \\ S_3 &= a_1 + a_2 + a_3 = 2+5+8=15 \\ S_4 &= S_3 + a_4= 2+5+8+11=26 \\ \vdots \end{aligned}

The partial sums above make up a new sequence $S_n = \{2,7,15,26, \dots\}$ . Determining the convergence or divergence of $S_n$ will be the focus of our work on infinite series.

Sigma Notation

Given a sequence of numbers $\{a_n\}_{n=1}^\infty$ , its infinite series can be written in sigma notation:

\sum_{i=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots + a_n + \cdots

The index is the variable $n$ (although variables $i$ and $k$ are commonly used as well). The lower bound is the starting point, and the upper bound is the stopping point. For a finite series, the upper bound is a finite value.

It is not possible to add an infinite number of terms by hand (unless every term is equal to zero), so the sum of the infinite series is defined as the limit of the sequence of partial sums $S_n$ as $n \to \infty$ :

S_\infty = \sum_{n=1}^\infty a_n = \lim_{n \to \infty} S_n.

For most series, calculating this limit is not possible. However, our focus is to determine whether the limit exists. If the limit exists, the series is said to converge. If the limit does not exist, the series is said to diverge.

The following are true regarding the sequence of partial sums $S_n$ :

$\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} S_n$
$a_n = S_n - S_{n-1}$
If a sequence $a_n$ converges, $S_n$ may or may not converge
If $S_n$ converges, the series $\sum_{n=1}^{\infty} a_n$ converges

Example 1

Given the partial sum $S_n = \frac{n-2}{n+2}$ , determine the general term $a_n$ .

View Answer

a_n = S_n - S_{n-1} = \frac{n-2}{n+2} - \frac{n-3}{n+1}.

Properties of Series

The following properties apply to series (with summation bounds being the same on both sides of the equation):

\begin{aligned} \sum (a_n \pm b_n) &= \sum a_n \pm \sum b_n \\ \sum c \cdot a_n &= c \sum a_n \end{aligned}

Note: These properties are similar to how integrals operate.

Example 2

Create two examples, one demonstrating each series property discussed above.

View Answer

Possible examples:

$\sum (3n + 2n) = \sum 3n + \sum 2n$
$\sum 5 \cdot n = 5 \sum n$

Rewriting Series

Sigma notation is not a unique representation of a series -- a series can be represented in a handful of ways. Often, it may be necessary to rewrite our series in order to work with it.

The two ways in which we rewrite series include index shifting and stripping out terms. Keep in mind that when you rewrite series in these two ways, it is only the appearance of the series that changes -- all the terms are exactly the same.

Index Shifting

Index shifting allows us to do just that -- shift the index of the series. For example:

\sum_{n=2}^\infty (-2)^{n-1} = \sum_{n=1}^\infty (-2)^n = \sum_{n=0}^\infty (-2)^{n+1}.

Note: If the index increases in $a_n$ , it decreases by the same amount in the summation bounds.

Example 3

Index shift $\sum_{n=2}^{7} (n-1)(n+1)$ to begin at $n=0$ and $n=3$ .

View Answer

$\sum_{n=0}^{5} ((n+2)-1)((n+2)+1)$
$\sum_{n=3}^{8} ((n-1)-1)((n-1)+1)$

Example 4

Index shift $\sum_{k=3}^{\infty} \frac{k!}{k-1}$ to begin at $k=2$ and $k=5$ .

View Answer

$\sum_{k=2}^{\infty} \frac{(k+1)!}{k}$
$\sum_{k=5}^{\infty} \frac{(k-2)!}{k-3}$

Stripping Out Terms

Stripping out terms allows us to do just that -- strip out terms from the beginning of the series. For example:

\begin{aligned} \sum_{n=0}^\infty \frac{1}{n+1} &= \frac{1}{1} + \sum_{n=1}^\infty \frac{1}{n+1}\\ \sum_{n=0}^\infty \frac{1}{n+1} &= \frac{1}{1} + \frac{1}{2} + \sum_{n=2}^\infty \frac{1}{n+1}\\ \sum_{n=0}^\infty \frac{1}{n+1} &= \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \sum_{n=3}^\infty \frac{1}{n+1} \end{aligned}

Not: The expression $a_n$ remains the same, but the starting point of the series changes.

Example 5

Strip out the first two terms of $\sum_{n=2}^\infty \sin n$ .

View Answer

\sin 2 + \sin 3 + \sum_{n=4}^\infty \sin n.

Example 6

Strip out the first three terms of $\sum_{n=1}^\infty a_n$ .

View Answer

a_1 + a_2 + a_3 + \sum_{n=4}^\infty a_n.

Convergence of a Series

When determining the convergence of a series, consider the following:

All finite series are convergent if each of their terms is a real number because a finite sum of real numbers is
If you add any real number to a real number, the sum is a real number
If you add any real number to infinity, the "sum" is infinity
Multiplying a real number by another real number results in a real number
Multiplying a real (non-zero) number by infinity results in infinity
If you strip out a finite number of terms from a series, it will not change the convergence or divergence of the series
If you add or subtract a finite number of terms from a series, it will not change the convergence or divergence of the series
If you multiply a series by a real number (other than zero), it will not change the convergence or divergence of the series
If you add two divergent series together, the result could be a convergent or divergent series

Example 7

If $\sum_{n=2}^\infty a_n$ and $\sum_{n=2}^\infty b_n$ are both convergent, what can be said about $\sum_{n=2}^\infty (a_n + b_n)$ ?

View Answer

The series $\sum_{n=2}^\infty (a_n + b_n)$ converges.

Example 8

If $\sum_{n=2}^\infty a_n$ and $\sum_{n=2}^\infty b_n$ are both divergent, what can be said about $\sum_{n=2}^\infty (a_n + b_n)$ ?

View Answer

The series $\sum_{n=2}^\infty (a_n + b_n)$ could converge or diverge.

Example 9

If $\sum_{n=2}^\infty a_n$ and $\sum_{n=2}^\infty b_n$ are both convergent, what can be said about $\sum_{n=2}^\infty (a_n - b_n)$ ?

View Answer

The series $\sum_{n=2}^\infty (a_n - b_n)$ converges.

Example 10

If $\sum_{n=4}^\infty a_n$ is convergent, what can be said about the following:

$\sum_{n=4}^\infty 10a_n$
$\sum_{n=4}^\infty -10a_n$
$\sum_{n=7}^\infty 10000000a_n$

View Answer

$\sum_{n=4}^\infty 10a_n$ converges.
$\sum_{n=4}^\infty -10a_n$ converges.
$\sum_{n=7}^\infty 10000000a_n$ converges.

Example 11

If $\sum_{n=1}^\infty b_n$ is divergent, what can be said about the following:

$\sum_{n=4}^\infty 10b_n$
$\sum_{n=4}^\infty 0.000002b_n$

View Answer

$\sum_{n=4}^\infty 10b_n$ diverges.
$\sum_{n=4}^\infty 0.000002b_n$ diverges.

Series Tests

Up to this point, we've discussed convergence of a series but we haven't discussed how to determine the convergence of a series.

To determine the convergence of a series, we utilize a handful of different tests:

nth Term/Divergence Test
Geometric Series Test
p-Series Test
Integral Test
Direct and Limit Comparison Tests
Alternating Series Test
Ratio Test
Root Test

These tests will be the focus of the next few lessons.

Alternating Series Error Bound

nth Term Test