Alternating series.

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

Example 9.4.1       

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

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

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$ ).

Noah Dana-Picard 2007-12-28