My Math Forum strange sum

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 September 29th, 2011, 12:12 PM #1 Member   Joined: Nov 2010 Posts: 77 Thanks: 0 strange sum $\sum _{n=1}^{\infty } \frac{(-1)^n \text{Cos}\left[\sqrt{b^2+n^2 \pi ^2}\right]}{n^2 \text{Binomial}[2 n,n]}$ some ideas I could simplify more $\left\{\frac{2\text{ArcCsch}[2]}{\sqrt{5}}-\frac{x^2\text{ArcCsch}[2]}{\sqrt{5}}+\frac{x^4\text{ArcCsch}[2]}{12\sqrt{5}}-\frac{1}{2}\text{HypergeometricPFQ}^{(\{0,0,0\},\{ 0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{4}x^2\text{HypergeometricPFQ}^{(\{0,0,0\ },\{0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{48}x^4\text{HypergeometricPFQ}^{(\{0,0,0\ },\{0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{4}\text{HypergeometricPFQ}^{(\{0,0,0\},\ {0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{8}x^2\text{HypergeometricPFQ}^{(\{0,0,0\} ,\{0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{12}\text{HypergeometricPFQ}^{(\{0,0,0\},\ {0,3\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{24}x^2\text{HypergeometricPFQ}^{(\{0,0,0 \},\{0,3\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{48}\text{HypergeometricPFQ}^{(\{0,0,0\}, \{0,4\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{240}\text{HypergeometricPFQ}^{(\{0,0,0\}, \{0,5\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{2}\text{HypergeometricPFQ}^{(\{0,0,1\},\{ 0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{4}x^2\text{HypergeometricPFQ}^{(\{0,0,1\ },\{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{48}x^4\text{HypergeometricPFQ}^{(\{0,0,1\ },\{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{2}\text{HypergeometricPFQ}^{(\{0,0,1\},\ {0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{4}x^2\text{HypergeometricPFQ}^{(\{0,0,1\} ,\{0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{4}\text{HypergeometricPFQ}^{(\{0,0,1\},\{ 0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{8}x^2\text{HypergeometricPFQ}^{(\{0,0,1\ },\{0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{12}\text{HypergeometricPFQ}^{(\{0,0,1\}, \{0,3\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{48}\text{HypergeometricPFQ}^{(\{0,0,1\},\ {0,4\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{4}\text{HypergeometricPFQ}^{(\{0,0,2\},\ {0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{8}x^2\text{HypergeometricPFQ}^{(\{0,0,2\} ,\{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{4}\text{HypergeometricPFQ}^{(\{0,0,2\},\{ 0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{8}x^2\text{HypergeometricPFQ}^{(\{0,0,2\ },\{0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{8}\text{HypergeometricPFQ}^{(\{0,0,2\},\ {0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{24}\text{HypergeometricPFQ}^{(\{0,0,2\},\ {0,3\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{12}\text{HypergeometricPFQ}^{(\{0,0,3\},\ {0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{24}x^2\text{HypergeometricPFQ}^{(\{0,0,3 \},\{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{12}\text{HypergeometricPFQ}^{(\{0,0,3\}, \{0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{24}\text{HypergeometricPFQ}^{(\{0,0,3\},\ {0,2\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]+\frac{1}{48}\text{HypergeometricPFQ}^{(\{0,0,4\}, \{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{48}\text{HypergeometricPFQ}^{(\{0,0,4\},\ {0,1\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]-\frac{1}{240}\text{HypergeometricPFQ}^{(\{0,0,5\}, \{0,0\},0)}\left[\{1,2,1\},\left\{\frac{3}{2},2\right\},-\frac{1}{4}\right]\}\right.$

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