<s>[INST] Series acceleration with Fourier-Bessel series coefficients I was investigating methods for series acceleration and I found this identity: 
$$e=\sum _{n=0}^{\infty } \left(\sqrt{\frac{\pi }{2}} (2 n+1)\right) I_{n+\frac{1}{2}}(1)$$
where $I$ is the modified Bessel function of the first kind.
Could you explain me where this identity comes from ? 
I tried computing the Fourier-Bessel series for the exponential function but it did not work.
 [/INST] good work Jack but the Integral it is very difficult to  resolve anyway i try to get the series of the Sinh(x)/x using series Bessel Fourier  and we get a different seriesthan you give it above i think the following is it faster
$$\frac{\sinh (x)}{x}=\sum _{n=0}^{\infty } I_{n+\frac{1}{2}}(1) \left(\pi  2^{n-\frac{7}{2}} \left((-1)^n+1\right) (2 n+1) x^n \left(\frac{(-1)^{\left\lfloor \frac{n}{2}\right\rfloor +1} 4^{-\left\lfloor \frac{n}{2}\right\rfloor } x^{-2 \left(\left\lfloor \frac{n}{2}\right\rfloor +1\right)} \sec \left(\pi  \left(n-\left\lfloor \frac{n}{2}\right\rfloor \right)\right) \, _3\tilde{F}_2\left(1,-\frac{n}{2}+\left\lfloor \frac{n}{2}\right\rfloor +\frac{1}{2},-\frac{n}{2}+\left\lfloor \frac{n}{2}\right\rfloor +1;\left\lfloor \frac{n}{2}\right\rfloor +2,-n+\left\lfloor \frac{n}{2}\right\rfloor +\frac{3}{2};\frac{1}{x^2}\right)}{\Gamma \left(n-2 \left\lfloor \frac{n}{2}\right\rfloor \right)}+\frac{4 \sec (\pi  n) \, _2\tilde{F}_1\left(-\frac{n}{2}-\frac{1}{2},-\frac{n}{2};\frac{1}{2}-n;\frac{1}{x^2}\right)}{\Gamma (n+2)}\right)\right)$$ but it is very complicate thanks anyway.
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