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Find a sequence $\{a_n\}$ of real numbers such that $\sum a_n$ converges but $\prod (1+ a_n)$ diverges.

The converse is trivial, just make all the $a_n=-1$.

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    Related: http://math.stackexchange.com/questions/119140/proof-of-a-theorem-of-cauchys-on-the-convergence-of-an-infinite-product2012-04-09

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It's not hard to show that if each $|a_n| < 1$ then $\prod_n(1 + a_n)$ converges to a nonzero value iff $\sum_n \log(1 + a_n)$ does (take logarithms of the partial products). If $a_n \rightarrow 0$ then for $n$ large enough, by Taylor expanding the log you have for example $ a_n -a_n^2 < \log(1 + a_n) < a_n - {a_n^2 \over 4}$ So if you take $a_n = (-1)^n{1 \over \sqrt{n}}$, even though $\sum {a_n}$ converges (it's a decreasing alternating series), the sum of the logarithms will diverge since $a_n^2 = {1 \over n}$. So the product will diverge as well.

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    Also I still don't see how you conclude that $\sum (-1)^n\frac1{\sqrt{n}}-\frac1n$ diverges. Every convergence test I've done is inconclusive.2012-04-09