Alternating Series - Alternating Series Test

Alternating Series Test

The theorem known as "Leibniz Test" or the alternating series test tells us that an alternating series will converge if the terms a_n converge to 0 monotonically.

Proof: Suppose the sequence converges to zero and is monotone decreasing. If is odd and, we obtain the estimate via the following calculation:

$\begin{align} S_m - S_n & = \sum_{k=0}^m(-1)^k\,a_k\,-\,\sum_{k=0}^n\,(-1)^k\,a_k\ = \sum_{k=m+1}^n\,(-1)^k\,a_k \\ & =a_{m+1}-a_{m+2}+a_{m+3}-a_{m+4}+\cdots+a_n\\ & =\displaystyle a_{m+1}-(a_{m+2}-a_{m+3}) - (a_{m+4}-a_{m+5}) -\cdots-a_n \le a_{m+1}\le a_{m} \end{align}$

Since is monotonically decreasing, the terms are negative. Thus, we have the final inequality .Similarly it can be shown that . Since converges to, our partial sums form a cauchy sequence (i.e. the series satisfies the cauchy convergence criterion for series) and therefore converge. The argument for even is similar.

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