Infinite Series

The Essentials

An infinite series is the sum of an infinite number of terms:

\[ \sum_{n=1}^\infty a_n=a_1+a_2+a_3+\ldots \]

The sum of a certain amount (k) of terms is written as \( S_k \) and is called a partial sum. The partial sums form a sequence: \( \{S_k\} \) . If the sequence of partial sums converges, then the series converges. The sum of a convergent series is S.

The harmonic series is an example of a series that diverges:

\[ \sum_{n=1}^\infty \frac{1}{n} \]

The series that is a sum or difference of convergent series converges. Constant multiples can be pulled out of a series. A geometric series has the form:

\[ a+ar+ar^2+ar^3+\ldots=\sum_{n=1}^\infty \]

An example of a geometric series is:

\[ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots \]

Where \( a=1 \) and \( r=1/2 \) . If \( |r|<0 \) , a geometric series converges with

\[ S=\frac{a}{1-r} \]

If \( |r|\geq 0 \) , the series diverges. A telescoping series is one where the middle terms cancel out and leave the first term and the last term. An example is:

\[ \sum_{n=1}^k\left[\frac{1}{n}-\frac{1}{n+1}\right] \]
\[ S_k=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\ldots+\left(\frac{1}{k}-\frac{1}{k+1}\right)=1-\frac{1}{k+1} \]

Example

Determine whether the series converges or diverges by using a sequence of partial sums. If it converges, find what it converges to:

\[ \sum_{n=1}^\infty \frac{1}{(n+2)(n+3)} \]

First, we are going to use partial fraction expansion to re-write the fraction:

\[ \sum_{n=1}^\infty \frac{1}{n+2}-\frac{1}{n+3} \]

The first couple of partial sums are:

\[ \begin{array}{ccccc} S_1 & = & 1/3-1/4\\ S_2 & = & 1/3-1/4+1/4-1/5 & = & 1/3-1/5\\ S_3 & = & 1/3-1/5+1/5-1/6 & = & 1/3-1/6 \end{array} \]

We can see that this is a telescoping series. If we start adding all of these terms together then until the kth term we will end up with:

\[ 1/3+1/k \]

When k finally reaches infinity, it will be 1/3. Therefore the series converges at 1/3.

Practice

  1. Using Sigma notation re-write this series:
    \[ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots \]
  2. Find the partial sum \(S_6 \) of the series:
    \[ a_n=n \]
  3. Use a sequence of partial sums to determine whether the series converges or diverges:
    \[ \sum_{n=1}^\infty \frac{n}{n+2} \]

Solutions:

  1. \[ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \]
  2. 21
  3. Diverges

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