# 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 be a sequence of positive real numbers. Suppose that:
• ;
• There exists a positive integer such that .
Then the alternating series is convergent.

Example 9.4.3 (The alternating harmonic series)   The series is convergent.

Theorem 9.4.4   Take a convergent alternating series , as in Theorem 4.2.
1. For , the partial sum is an approximation of the sum of the series, with an error which is less than .
2. The remainder and the first unused term have the same sign.

Example 9.4.5   Take the alternating harmonic series . Then

is an approximation of the sum of the series with an error less than .

In order to have an approximation of the sum of the series with an error less than , we must add the first 100 terms of the series (we get ).

Noah Dana-Picard 2007-12-28