Series

MATH1051

A finite series is a sum of finitely many terms $a_1+a_2+a_3+\cdots +a_n$. An infinite series is a sum of infinitely many terms $a_1+a_2+a_3+\dots$

We shall see that if we add an infinite number of terms the result may be finite or infinite.

If the series has a finite sum, we say it converges.

Whether or not a series converges is not obvious.

7.1 Infinite Sums (notation)

If we have an infinite sum we write:

$$a_0+a_1+a_2+\cdots +a_n+\cdots =\sum _{n=0}^{\infty }{a}_n.$$

Note that the lower bound ($n=0$) of the sum may vary.

7.2 Motivation

Series come from many fields.

1. Approximation to problem solutions:
   1. $a_0$ zeroth order approximation
   2. $a_0+a_1$ ($a_1$ small) first order approximation
   3. $a_0+a_1+a_2$ ($a_2$ very small) second order approximation
   4. $a_0+a_1+a_2+\cdots +a_n$ $n$th order
   5. $a_0+a_1+a_2+\cdots +a_n+\dots$ exact solution, provided the series converges
2. Current state of a process over infinite time horizon
3. Approximating functions via Taylor/Fourier Series, e.g.,

$$e^x=1+e^x+\frac{1}{2}{x}^2+\frac{1}{3!}{x}^3+\dots$$ $$f(t)=c_0+c_1\sin t+c_2\sin (2t)+c_3\sin (3t)+\dots$$

4. Riemann Sums

7.3 The Harmonic Series

The series $1+\frac{1}{2}+\frac{1}{3}+\cdots ={\sum }_{n=1}^{\infty }\frac{1}{n}$ is called the Harmonic series

7.4 Definition of Convergence

Given a series ${\sum }_{n=0}^{\infty }{a}_n=a_0+a_1+a_2+\dots$, let $s_n$ denote its $n$th partial sum:

$$\begin{array}{rl}{s}_0& =a_0\\ s_1& =a_0+a_1\\ s_2& =a_0+a_1+a_2\\ ⋮& \\ s_n& =a_0+a_1+\cdots +a_n=\sum _{k=0}^n{a}_k\end{array}$$

If the sequence $\{s_n\}$ is convergent (i.e. ${\lim }_{n\to \infty }{s}_n=s$ with $s\in \mathbb{R}$), then the series ${\sum }_{n=0}^{\infty }{a}_n$ is said to be convergent and we write

$$\underset{n\to \infty }{\lim }{s}_n=\sum _{n=0}^{\infty }{a}_n=s.$$

The number $s$ is called the sum of the series. Otherwise the series is said to be divergent.

7.5 The P-test

For $p\in \mathbb{R}$, the $p-$series ${\sum }_{n=1}^{\infty }\frac{1}{n^p}$ is convergent if $p>1$ and divergent if $p\le 1$

Note the above sum is from $n=1$. This is just a matter of taste since

$$\sum _{n=1}^{\infty }\frac{1}{n^p}=\sum _{n=0}^{\infty }\frac{1}{(n+1{)}^p}$$

7.6 The Divergence Test (also Called the Nth Term test)

If ${\sum }_{n=0}^{\infty }{a}_n$ is convergent then ${\lim }_{n\to \infty }{a}_n=0.$

The divergence test is if ${\lim }_{n\to \infty }{a}_n\ne 0$ then the series is divergent.

7.7 Geometric Series

The series

$$\sum _{n=0}^{\infty }a{r}^n$$

is convergent if $|r|<1$ with ${\sum }_{n=0}^{\infty }a{r}^n=\frac{a}{1-r}$ and divergent if $|r|\ge 1$.

7.8 Application: Bouncing Ball

Error

Please look at the workbook online for the notes. Takes too long to write with all the working

7.9 The Comparison Test

If we are trying to determine whether or not ${\sum }_{n=0}^{\infty }{a}_n$ converges, where $a_n$ contains complicated terms, then it may be possible to bound the series by using simpler terms. If $|a_n|$ is always smaller than $|b_n|$ and ${\sum }_{n=0}^{\infty }|b_n|$ converges, then ${\sum }_{n=0}^{\infty }{a}_n$ should also. On the other hand, if $a_n$ is always large - in fact larger than $|b_n|$, and ${\sum }_{n=0}^{\infty }{b}_n$ diverges, then ${\sum }_{n=0}^{\infty }{a}_n$ should diverge too.

7.10 Alternating Series

An alternating series is a series whose terms alternate in signs.

Here is an interesting result about a special alternating series:

We have seen that the harmonic series

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots$$

is divergent. However, the alternating series

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$$

is convergent. Why? Later, we shall see that this series converges to $\ln 2$.

If the alternating series

$$\sum _{n=0}^{\infty }(-1{)}^n{b}_n=b_0-b_1+b_2-b_3+\dots (\text{all }{b}_n>0)$$

satisfies

$$\underset{n\to \infty }{\lim }{b}_n=0\text{ and }$$ $$b_{n+1}\le b_n,\text{ for all }n\ge 0$$

then the series is convergent. The alternating series test can only be used to show convergence, not divergence.

7.11 Absolute and Conditional Convergence

A series ${\sum }_{n=0}^{\infty }{a}_n$ is called absolutely convergent if the series ${\sum }_{n=0}^{\infty }|a_n|$ is convergent. A series ${\sum }_{n=0}^{\infty }{a}_n$ is called conditionally convergent if it is convergent, but not absolutely convergent.

7.11.1 Example

Are the following series conditionally or absolutely convergent?

$$\sum _{n=1}^{\infty }(-1{)}^n\frac{1}{n}$$

${\sum }_{n=1}^{\infty }(-1{)}^n\frac{1}{n}$ is an alternating series with $b_n=\frac{1}{n}$. Since $b_n>b_{n+1}$, for all $n\ge 1$, and ${\lim }_{n\to \infty }{b}_n=0$, then by the alternating series test, this series converges.

Now let $a_n=(-1{)}^n\frac{1}{n}$ for $n\ge 1$. Then

$$\sum _{n=1}^{\infty }|a_n|=\sum _{n=1}^{\infty }\frac{1}{n},$$

and this series is divergent ($p$-series with $p=1$.)

Consequently, the series is conditionally convergent.

7.12 The Ratio Test

A powerful test for convergence of series is the ratio test. We will use it extensively in the following sections.

If ${\lim }_{n\to \infty }\mid \frac{a_{n+1}}{a_n}\mid =L<1$, then the series ${\sum }_{n=1}^{\infty }{a}_n$ is absolutely convergent (and therefore convergent).

If ${\lim }_{n\to \infty }\mid \frac{a_{n+1}}{a_n}\mid =L>1$ or ${\lim }_{n\to \infty }\mid \frac{a_{n+1}}{a_n}\mid =\infty$, then the series ${\sum }_{n=1}^{\infty }{a}_n$ is divergent.

If ${\lim }_{n\to \infty }\mid \frac{a_{n+1}}{a_n}\mid =1$, then the ratio test is inconclusive.

If ${\lim }_{n\to \infty }\mid \frac{a_{n+1}}{a_n}\mid$ is not defined, then the ratio test is inconclusive.