Let denote the Bessel function of first kind. Evaluate the series:

**Solution**

We recall the fact that

(1)

as well as the Fourier series

(2)

Hence,

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Let denote the Bessel function of first kind. Evaluate the series:

**Solution**

We recall the fact that

(1)

as well as the Fourier series

(2)

Hence,

Some comments and extensions as well:

Remark 1: Let then it holds that:Remark 2: Let then it holds that:Remark 3: Let be a real parameter such that . Then it holds that: