Alternating series.

An alternating series is a series where the terms are alternatively positive and negative.

Example 9.4.1       

• $\underset{n=1}{\overset{+\infty }{\sum}} \frac {(-1)^{n+1}}{n} = 1 - \frac 12 + \frac 13 - \frac 14 + \frac 15 - \dots$ .
• $\underset{n=0}{\overset{+\infty }{\sum}} \frac {(-1)^{n}}{n} = \frac {1}{0!} -... ... {1}{1!} +\frac {1}{2!} - \frac {1}{3!} + \frac {1}{4!} - \frac {1}{5!} + \dots$ .

Theorem 9.4.2 (Leibniz)   Let $(u_n)_{n \geq 0}$ be a sequence of positive real numbers. Suppose that:

• $\underset{n \rightarrow + \infty }{\lim} u_n = 0$ ;
• There exists a positive integer $N_0$ such that $\forall n \geq N_0, u_{n+1} > u_n$ .

Then the alternating series $\underset{n=0}{\overset{+\infty }{\sum}} (-1)^{n+1} u_n$ is convergent.

Example 9.4.3 (The alternating harmonic series)   The series $\underset{n=1}{\overset{+\infty }{\sum}} \frac {(-1)^{n+1}}{n}$ is convergent.

Theorem 9.4.4   Take a convergent alternating series $\underset{k=0}{\overset{+\infty }{\sum}} (-1)^{k+1} u_k$ , as in Theorem 4.2.

1. For $n \geq N_0$ , the $n^{th}$ partial sum $S_n= \underset{k=0}{\overset{n}{\sum}} (-1)^{k+1} u_k$ is an approximation of the sum $S$ of the series, with an error which is less than $\vert u_{n+1}\vert$ .
2. The remainder $S-S_n$ and the first unused term $u_{n+1}$ have the same sign.

Example 9.4.5   Take the alternating harmonic series $\underset{n=1}{\overset{+\infty }{\sum}} \frac {(-1)^{n+1}}{n}$ . Then

$$S_{10}=1-\frac 12 + \frac 13 - \frac 14 + \frac15 - \frac 16 + \frac 17 - \frac 18 - \frac 19 + \frac {1}{10} = \frac {1627}{2520}$$

is an approximation of the sum of the series with an error less than $\frac {1}{11} = 0.090909...$ .

In order to have an approximation of the sum of the series with an error less than $10^{-2}$ , we must add the first 100 terms of the series (we get $S \approx 0.68817$ ).