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I am trying to show $$\sum_{n=0}^\infty (-1)^n\frac{1\cdot3 \cdots (2n-1)}{2\cdot 4 \cdots (2n)}$$ is conditionally convergent. If I can show $a_n \rightarrow 0$ then I can apply alternating series test to show convergence. I am trying to avoid the use of Stirling's formula. I do not know how to attack divergence of $\sum |a_n| $ with out employing Stirling's formula.

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    $\begingroup$ It is elementary to show that $\frac{1}{2\sqrt{n}}\leq |a_n|\leq \frac{1}{\sqrt{2n+1}}$. I do not think this question is about research level mathematics. $\endgroup$ – abatkai Aug 2 '13 at 21:57
  • $\begingroup$ Specifically, abatkai's inequalities may be proved by an inductive argument. -- Todd Trimble $\endgroup$ – user43208 Aug 4 '13 at 2:02
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We have $$\log |a_n|=\sum_{j=1}^n\log\left(1-\frac 1{2j}\right).$$ Now compare $\log(1-x)$ with $x$ in order to find a below bound for $\log |a_n|$.

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