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 an 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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