# Show the convergence of a series

Given the series:

$$S(\lambda,\Phi)=\sum_{n=0}^{\infty}{J_n}(\lambda)e^{in\Phi}$$

where the $J_n(\lambda)$ is the Bessel function of order $n$

I have some difficulty to give a proof of its convergence. Can someone help me?

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This series is well-known in the theory of the Bessel functions and Fourier series being the representation of

$$S(\lambda,\Phi)=\sum_{n=0}^\infty J_n(\lambda)e^{in\Phi}=e^{i\lambda\sin\Phi}$$

and this Fourier series is converging as can be seen from the fact that

$$\left|\sum_{n=0}^\infty J_n(\lambda)e^{in\Phi}\right|<\sum_{n=0}^\infty |J_n(\lambda)|<\infty.$$

for $\lambda$ finite.

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If you don't mind me asking, how did you know that $\sum_{n=0}^{\infty} |J_{n}(\lambda)| < \infty$ for finite $\lambda$? I was only able to prove it for $|\lambda|<2$. –  Nick Thompson Feb 14 '12 at 11:29
@NickThompson: Yours is a good question but I know the sum of the series that is defined for any finite $\lambda$. You can also wok this out through the coefficients of the Fourier series $$c_n=\frac{1}{2\pi}\int_0^{2\pi}e^{i\lambda\sin\Phi-in\Phi}d\Phi$$. –  Jon Feb 14 '12 at 11:36