Math 2565, Spring 2020

Sequences and Series

Sean Fitzpatrick
University of Lethbridge

Warm-Up and Quiz

Warm-Up

Find the limit of the sequence:

  1. \(\di \lim_{n\to\infty}\frac{2^n}{3^n}\)

  2. \(\di \lim_{n\to\infty}(\ln(n+1)-\ln(n))\)

Quiz

Decide if the following geometric series converge.

  1. \(\displaystyle \sum_{n=0}^\infty \left(\frac35\right)^n\)

  2. \(\displaystyle \sum_{n=0}^\infty \left(\frac53\right)^n\)

Sequences, continued

Properties of sequences

Limit properties are pretty much what you expect. Assume \(\{a_n\}\) and \(\{b_n\}\) converge. Then:

  • \(\di \lim_{n\to\infty}(a_n\pm b_n) = \lim_{n\to \infty}a_n \pm \lim_{n\to\infty}b_n\)

  • \(\di \lim_{n\to \infty}ca_n = c\lim_{n\to \infty}a_n\)

  • \(\di \lim_{n\to \infty}(a_nb_n) = \lim_{n\to \infty}a_n\cdot\lim_{n\to\infty}b_n\)

  • If \(f\) is continuous, \(\lim\limits_{n\to \infty}f(a_n)=f\left(\lim\limits_{n\to\infty}a_n\right)\text{.}\)

Bounded and monotone sequences

Definition:

A sequence \(\{a_n\}\) is bounded if \(m\lt a_n \lt M\) for some real \(m,M\text{.}\)

A sequence is monotone increasing if \(a_n\lt a_{n+1}\) for all \(n\text{.}\)

A sequence is monotone decreasing if \(a_n\gt a_{n+1}\) for all \(n\text{.}\)

Theorem:

  1. Any convergent sequence is bounded.

  2. Any bounded, monotone sequence converges.

Examples

Of the following sequences, which are bounded? Which are montone?

  1. \(\di a_n = \frac{5}{2n+1}\)

  2. \(\di a_n = \frac{2n-1}{3n+5}\)

  3. \(\di a_n = \sin(n\pi/4)\)

  4. \(\di a_n = \frac{n^2-4}{3n}\)

Example

Let \(\{a_n\}\) be defined as follows:

\begin{align*} a_1 \amp = \sqrt{3}\\ a_{n+1} \amp =\sqrt{3a_n}, \text{ for } n\geq 1\text{.} \end{align*}

Show that \(\{a_n\}\) converges, and find its limit.

Series

Partial sums

Given a sequence \(\{a_n\}\text{,}\) construct a new sequence \(\{s_n\}\) by

\begin{align*} s_1 \amp = a_1 \\ s_2 \amp = a_1 + a_2\\ s_3 \amp = a_1+a_2+a_3 \end{align*}
and so on.

In general, \(s_n = \sum_{k=1}^n a_k\text{.}\)

Series

A series is the limit of a sequence of partial sums. We write

\begin{equation*} \sum_{n=1}^\infty a_n = \lim_{n\to\infty}s_n\text{,} \end{equation*}
where \(\{s_n\}\) is the sequence of partial sums.

If the limit exists (and is finite) we say the series converges. Otherwise, it diverges.

Example:

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

Geometric series

Geometric series have the form

\begin{equation*} \sum_{n=0}^\infty ar^n\text{,} \end{equation*}
where \(a\) and \(r\) are real numbers.

Question: when does the series converge?

Variation:

\begin{equation*} \sum_{n=k}^\infty ar^n\text{.} \end{equation*}

\(p\)-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 \(\ln(2)\))!

Telescoping series

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

Example:

Determine the converge of:

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

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

Properties

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

  1. \(\di \sum(a_n\pm b_n) = \sum a_n \pm \sum b_n\)

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

Example:

Evaluate the sums

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

  2. \(\di\frac18+\frac{1}{16}+\frac{1}{32}+\cdots\)

  3. \(\di \frac19-\frac{1}{16}+\frac{1}{25}-\frac{1}{36}-\cdots\)