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:
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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