Series Convergence and Divergence

Section 3.16 Series Convergence and Divergence

Subsection 3.16.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

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Subsection 3.16.2 Introduction

An infinite series is a summation of the form

\begin{equation*} \sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \dots\text{.} \end{equation*}

A common type of infinite series is the geometric series, where each term in the series is obtained by multiplying the previous term by a fixed ratio, such as

\begin{equation} \sum_{n=0}^{\infty} \frac{1}{2^n} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\text{.}\tag{3.17} \end{equation}

To understand the result of the sum in equation (3.17), we can add the first few terms, called a partial sum. Looking at a sequence of partial sums, one at a time, may give an indication of the sum of the infinite series.

\begin{align*} 1+\frac{1}{2} \amp = \frac{3}{2}\\ 1+\frac{1}{2}+\frac{1}{4} \amp = \frac{7}{4}\\ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \amp = \frac{15}{8}\\ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n} \amp = \frac{2^{n+1}-1}{2^n} \end{align*}

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By looking at these partial sums, we can see that the sum is approaching the value \(2\text{.}\) We say that this series is convergent. In other cases, the partial sums do not approach a finite value. We say that these series are divergent.

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In this activity, you will use Maple’s Sum() command to quickly evaluate whether a series is convergent or divergent. Maple will give the value of the sum for a convergent series and will give the value \(\infty\) or \(-\infty\) if the series is divergent.

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Exercises 3.16.3 Exercises

1.

For each of the following series, set up the series symbolically using the Sum() command. Then, use the value(%) command to evaluate the sum and determine if it converges.

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(a)

\(\displaystyle\sum_{n=0}^{\infty} \dfrac{4^n}{n!}\)

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Aside

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(b)

\(\displaystyle\sum_{n=0}^{\infty} \sin(n\pi)\arctan(n)\)

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Hint.

Make sure to use Pi for \(\pi\) (or use the palettes toolbar) and place multiplication between the \(n\) and \(\pi\text{.}\)

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(c)

\(\displaystyle\sum_{n=1}^{\infty} \ln(n)\)

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(d)

\(\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^2}\)

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2.

Another method for determining convergence or divergence of an infinite series is by comparing the series to a improper integral. This only applies to very specific infinite series.

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The Integral Test for Convergence states that for a non-negative, monotonically decreasing function \(f(n)\) and an integer \(N\text{,}\) the infinite series

\begin{equation*} \sum_{n=N}^{\infty} f(n) \end{equation*}

converges to a real number if and only if the improper integral

\begin{equation*} \int_N^{\infty} f(x)\,dx \end{equation*}

is finite. From this, we can also conclude that if the integral diverges, then the series diverges as well.

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In this exercise you will determine the convergence or divergence of the series

\begin{equation*} \sum_{n=3}^{\infty}\dfrac{3}{n^2-3n+2} \end{equation*}

using the Integral Test.

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(a)

Graph the function \(f(x) = \dfrac{3}{x^2-3x+2}\) to see that the function is non-negative and monotonically decreasing over the interval \([3,\infty)\text{.}\)

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Hint.

After plotting \(f(x)\text{,}\) try to think about why this series starts at \(N=3\text{.}\)

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(b)

Evaluate the integral given in the Integral Test to determine whether the series converges or diverges.

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