Just as the topic ask how to evaluate $\sum_{k=1}^{\infty}\frac{(18)[(k-1)!]^2}{(2k)!}.$
Evaluate $\sum\limits_{k=1}^{\infty}\frac{(18)[(k-1)!]^2}{(2k)!}$
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real-analysis
sequences-and-series
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0possible duplicate of [How to prove by arithmetical means that $\sum\limits_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum\limits_{k=1}^{\infty}\frac{1}{k^{2}}$](http://math.stackexchange.com/questions/99809/how-to-prove-by-arithmetical-means-that-sum-limits-k-1-infty-frack-1) – 2012-06-18
2 Answers
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Notice that $ \frac{(k-1)!^2}{(2k)!} = \frac{\Gamma(k) \Gamma(k)}{ \Gamma(2k+1) } = \frac{1}{2k} B(k,k) = \frac{1}{2k} \int_0^1 x^{k-1} (1-x)^{k-1} \mathrm{d} x $ Thus $ \begin{eqnarray} \sum_{k=1}^\infty \frac{(k-1)!^2}{(2k)!} &=& \int_0^1 \left( \sum_{k=1}^\infty \frac{1}{2k} x^{k-1} (1-x)^{k-1} \right) \mathrm{d} x = \int_0^1 \frac{-\ln(1-x+x^2)}{2x (1-x)} \mathrm{d} x \\ &=& -2 \int_{-1/2}^{1/2} \frac{\ln(3/4+u^2)}{1-4 u^2} \mathrm{d} u = -2 \int_0^1 \frac{\ln((3 +u^2)/4)}{1-u^2} \mathrm{d} u = \frac{\pi^2}{18} \end{eqnarray} $
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In order to fill in on the last equality, define $ f(t) = -2 \int_0^1 \frac{\ln\left(1 - t(1-u^2) \right)}{1-u^2} \mathrm{d} u $ Clearly $f(0) = 0$, and we are interested in computing $f\left(\frac{1}{4} \right)$. $ f^\prime(t) = 2 \int_0^1 \frac{\mathrm{d} u}{1 + t(1-u^2)} \stackrel{{u = \sqrt{\frac{1-t}{t}} \tan(\phi)}}{=} \int_0^{\arcsin(\sqrt{t})} \frac{2 \mathrm{d} \phi}{\sqrt{t(1-t)}} = \frac{2 \arcsin(\sqrt{t})}{\sqrt{t(1-t)}} = \\ 2 \frac{\mathrm{d}}{\mathrm{d} t} \arcsin^2(\sqrt{t}) $ Thus $ f\left(\frac{1}{4} \right) = \int_0^{\frac{1}{4}} 2 \frac{\mathrm{d}}{\mathrm{d} t} \arcsin^2(\sqrt{t}) = 2 \arcsin^2\left(\frac{1}{2}\right) = \frac{\pi^2}{18} $
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0@Sasha: that does look similar mod a change of variables. We also dropped to the derivative at different points. (+1) – 2012-04-06
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Maple gives the answer as $36 \arcsin(1/2)^2$. More generally, $ \sum_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} t^k = 2 \arcsin \left(\frac{\sqrt{t}}{2}\right)^2$
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0@Mathematics: I derive an equivalent series in [this answer](http://math.stackexchange.com/questions/99809/how-to-prove-by-arithmetical-means-that-sum-limits-k-1-infty-frack-1/128680#128680). – 2012-04-06