Apr 1st, 2015
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PROOFLet us presume that converges absolutely, so converges. First Step is a series with nonnegative terms. If , then , and . If , then (because a positive number has to be be greater than a negative number), and so . Second step. converges. If we consider cases as in Step 1, I have . Adding to both sides, I will have. The series converges by assumption, so converges as well. Therefore, the inequality shows that converges by comparison. Third Step converges. Since converges, its negative converges. I have Since the two series on the right side of the equation converge, we now un

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