Expressing trigonometric function in terms of integral of Bessel function
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Observe that $2(n+1)(2\,n+1)!=(2\,n+2)!$. Then $$ \sum_{n=0}^\infty \frac{(1)^nx^{2n+1}}{2(n+1)(2n+1)!}=\frac1x\sum_{n=0}^\infty \frac{(1)^nx^{2n+2}}{(2\,n+2)!}. $$
Comments

Shibli over 3 years
I am trying to show that,
\begin{align*} \frac{1\cos x}{x} = \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta \end{align*}
I did the following but cannot figure out how to continue.
\begin{align*} \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta &=\sum_{n=0}^\infty \frac{(1)^nx^{2n+1}}{2(n+1)(n!)^24^n} \int_{0}^{\pi/2}\cos^{2n+1}\theta\,d\theta\cr &=\sum_{n=0}^\infty \frac{(1)^nx^{2n+1}}{2(n+1)(n!)^24^n}\cdot \frac{(n!)^24^n}{(2n+1)!}\cr &=\sum_{n=0}^\infty \frac{(1)^nx^{2n+1}}{2(n+1)(2n+1)!} \end{align*}

Shibli over 7 yearsSo you mean that the summation of series is $1\cos x$. I googled but could not find a reference about the expansion of $1\cos x$. Could you refer me to a document?

Julián Aguirre over 7 yearsYou know the series for the Bessel function but you do not know the one for $\cos x$?