Math 2565, Spring 2020

Convergence Tests

March 5, 2020

Sean Fitzpatrick
University of Lethbridge

Introduction and recap

Warm-Up

Decide if the series converges:

  1. \(\di \sum_{n=0}^\infty \frac{5n^7-3n^4+19n^3+2565}{11n^{12}+32n^9+55n^7+2020n^4+3}\)

  2. \(\di \sum_{n=1}^\infty \frac{n^2}{2^n}\)

  3. \(\di \sum_{n=1}^\infty \frac{n!}{n^n}\)

Quiz 10

Decide if the series converges:

  1. \(\di \sum_{n=0}^\infty \frac{3^n}{2^n}\)

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

Root test

Theorem:

Let \(\{a_n\}\) be a sequence with positive terms, and consider \(\lim_{n\to \infty}\sqrt[n]{a_n}\text{.}\)

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

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

  2. \(\sum_{n=1}^\infty \frac{1}{n^n}\)

Alternating Series

Alternating Series Test

An alternating series is a series of the form \(\di \sum_{n=1}^\infty (-1)^na_n\text{.}\)

Theorem:

If \(\{a_n\}\) is a positive, decreasing sequence with \(\lim\limits_{n\to\infty}a_n=0\text{,}\) then \(\di \sum_{n=1}^\infty (-1)^na_n\) converges.

Examples: \(\di \sum_{n=1}^\infty \frac{(-1)^n}{n}\text{,}\) \(\di\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}\)

Absolute convergence

Most of our other convergence tests apply to series with positive terms. To apply the ratio test, etc. to an alternating series, take absolute values.

Definition ():

A series \(\sum a_n\) converges absolutely if \(\sum \abs{a_n}\) converges.

If \(\sum a_n\) converges but \(\sum \abs{a_n}\) does not, we say that the series converges conditionally.

Theorem:

Any absolutely convergent series is convergent.

Approximation Theorem

In an alternating series, the partial sums jump back and forth across the limit.

Each time the jumps get smaller (why?)

Result: let \(L = \lim\limits_{N\to\infty} s_N = \sum_{n=1}^\infty (-1)^na_n\text{.}\) Then

\begin{equation*} \abs{S_N - L} \leq S_{N+1} \end{equation*}
for each \(N\geq 1\text{.}\)

Example: \(\di \sum_{n=1}^\infty \frac{(-1)^n}{n^4} = \frac{\pi^4}{720}\text{.}\) How many terms do we need to take in the series to get an answer accurate to four decimal places?

Power Series

Power series

A power series is a function of the form \(f(x) = \sum_{n=0}^\infty a_n x^n\text{.}\)

The domain for such a function is the set of all \(x\) for which the series converges.

Can also consider power series centred at some number \(c\text{:}\)

\begin{equation*} \sum_{n=0}^\infty a_n(x-c)^n\text{.} \end{equation*}

Examples

  1. \(\di \sum_{n=0}^\infty 2^nx^n\)

  2. \(\di\sum_{n=0}^\infty\frac{(-1)^n}{5^n}(x-3)^n\)

  3. \(\di \sum_{n=0}^\infty \frac{3^nx^n}{n^2}\)

Note that the first two series are geometric. What's the sum?

Radius and interval of convergence

Applying the ratio test to \(\di\sum_{n=0}^\infty a_n(x-c)^n\) produces the limit

\begin{equation*} \lim_{n\to\infty}\frac{\abs{a_{n+1}(x-c)^{n+1}}}{\abs{a_n(x-c)^n}} = \lim_{n\to\infty}\abs{\frac{a_{n+1}}{a_n}}\abs{x-c}\text{.} \end{equation*}

We need this limit to be less than 1.

So we need \(\di \abs{x-c}\lt 1/L\text{,}\) where \(\di L = \lim_{n\to\infty}\abs{\frac{a_{n+1}}{a_n}}\text{.}\)

Define \(R=1/L\) as the radius of convergence. (\(R=0\) if \(L=\infty\) and \(R=\infty\) if \(L=0\text{.}\))

Examples

Determine radius and interval of convergence:

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

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

  3. \(\di \sum_{n=0}^\infty \frac{(-1)^n}{n^3}x^n\)

  4. \(\di \sum_{n=0}^\infty \frac{(2x)^n}{n!}\)

Derivatives and Integrals

Did we get to this before time ran out? Let's do some examples on the board!