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Math 2565, Spring 2020

Convergence Tests

March 5, 2020

Sean Fitzpatrick
University of Lethbridge

Introduction and recap

Warm-Up

Decide if the series converges:

\(\di \sum_{n=0}^\infty e^{-n}\)
\(\di \sum_{n=1}^\infty \frac{1}{\sqrt{n}}\)
\(\di \sum_{n=1}^\infty \frac{1}{n^2}\)

Telescoping series

Sometimes, terms in a series cancel (if we're lucky).

Example:

Determine the converge of:

\(\di \sum_{n=1}^\infty\frac{1}{n^2+n}\)
\(\di \sum_{n=4}^\infty \frac{1}{n^2-3n}\)

Properties

Let \(\sum a_n\) and \(\sum b_n\) be convergent series. Then:

\(\di \sum(a_n\pm b_n) = \sum a_n \pm \sum b_n\)
For any constant \(c\text{,}\) \(\di\sum ca_n = c\sum a_n\)

Example:

Evaluate the sums

\(\di \sum_{n=2}^\infty\left(\frac{1}{4^n}-\frac{3}{n^2}\right)\)
\(\di\frac18+\frac{1}{16}+\frac{1}{32}+\cdots\)
\(\di \frac19-\frac{1}{16}+\frac{1}{25}-\frac{1}{36}-\cdots\)

Integral Test

Integral test

The integral test lets us use what we already learned from improper integrals:

Theorem:

Let \(f\) be a positive, continuous, decreasing function on \([1,\infty)\text{,}\) and let \(\{a_n\} = \{f(n)\}\text{.}\) Then \(\di\sum_{n=1}^\infty a_n\) converges if and only if \(\di\int_1^\infty f(x)\,dx\) converges.

Example: \(\di \sum_{n=1}^\infty \frac{1}{n\ln(n)}\)

\(p\)-series

A \(p\)-series is a series of the form

\begin{equation*} \sum_{n=1}^\infty \frac{1}{n^p}\text{.} \end{equation*}

A \(p\)-series:

converges, if \(p>1\)
diverges, if \(p\leq 1\)

A famous \(p\)-series is \(\di \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}\text{.}\)

Harmonic series

Theorem: Test for Divergence.

If the series \(\di\sum_{n=1}^\infty a_n\) converges, then \(\di\lim_{n\to\infty}a_n=0\text{.}\)

Example: Harmonic series.

The series \(\di\sum_{n=1}^\infty \frac1n\) is called the harmonic series.

Despite the fact that \(\frac1n\to 0\) as \(n\to\infty\text{,}\) the harmonic series diverges.

However, the alternating harmonic series \(\di \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\) converges (to \(\ln(2)\))!

Estimation

Theorem:

Let \(\{a_n\}=\{f(n)\}\) be a sequence with positive terms for which the integral test applies, and \(\sum a_n\) converges. Let

\begin{equation*} R_n = \sum_{n+1}^\infty a_n = \sum_{n=1}^\infty a_n - s_n, \end{equation*}

where \(\{s_n\}\) is the sequence of partial sums. Then

\begin{equation*} \int_{n+1}^\infty f(x)\,dx \leq R_n\leq \int_n^\infty f(x)\,dx\text{.} \end{equation*}

Example:

Approximate \(\sum_{n=1}^\infty\frac{1}{n^4}\) using the first 5 terms.

Comparison tests

Direct comparison

Works the same as comparison for improper integrals.

Theorem:

Let \(\{a_n\},\{b_n\}\) be sequences with positive terms, such that \(a_n\leq b_n\) for all \(n\text{.}\)

If \(\sum a_n\) diverges, then \(\sum b_n\) diverges.
If \(\sum b_n\) converges, then \(\sum a_n\) converges.

Examples

\(\di\sum_{n=1}^\infty \frac{1}{n^2+n}\)
\(\di \sum_{n=1}^\infty \frac{1}{\sqrt{2n^2-1}}\)
\(\di\sum_{n=2}^\infty \frac{1}{n^3-1}\)

Limit comparison

Theorem:

Let \(\{a_n\},\{b_n\}\) be sequences with positive terms, such that \(a_n\leq b_n\) for all \(n\text{.}\) If

\begin{equation*} \lim_{n\to\infty}\frac{a_n}{b_n}=L\text{,} \end{equation*}

where \(0\lt L\lt \infty\text{,}\) then \(\sum a_n\) and \(\sum b_n\) both converge, or both diverge.

Example:

Determine the convergence of

\(\sum_{n=1}^\infty\frac{3n^3-7n^2+5}{9n^7+8n^4+2565}\)
\(\sum_{n=1}^\infty \frac{1}{n+\ln(n)}\)

Ratio and Root Tests

Ratio test

Theorem:

Let \(\{a_n\}\) be a sequence with positive terms. Consider

\begin{equation*} \lim_{n\to\infty}\frac{a_{n+1}}{a_n}\text{.} \end{equation*}

If this limit is less than 1, the series \(\sum_{n=1}^\infty a_n\) converges. If it is greater than 1, the series diverges.

Example:

Determine the convergence of

\(\di\sum_{n=1}^\infty\frac{2^n}{n!}\)
\(\di\sum_{n=1}^\infty \frac{1}{n^2+4n}\)

More examples

\(\di\sum_{n=1}^\infty \frac{3n!}{(3n)!}\)
\(\di \frac{2\sqrt{2}}{9801}\sum_{n=1}^\infty \frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}}\)

Root test

Theorem:

Let \(\{a_n\}\) be a sequence with positive terms, and consider

\begin{equation*} \lim_{n\to \infty}\sqrt[n]{a_n}\text{.} \end{equation*}

If this limit is less than 1, then \(\sum a_n\) converges. If the limit is greater than 1, the series diverges.

Example:

Determine the convergence of

\(\sum_{n=1}^\infty \left(\frac{3n+5}{5n+3}\right)\)
\(\sum_{n=1}^\infty \frac{1}{n^n}\)

Math 2565, Spring 2020 Convergence Tests March 5, 2020 Sean Fitzpatrick University of Lethbridge