Consider series of the form $S=\sum_{n\ge1}f(n)P(n)$, where $f$ is some smooth function, and $P$ is a periodic or quasi-periodic function (e.g., $P$ can be a trigonometric function, so $S$ a Fourier series). I would like to compute $S$ numerically to a reasonably large number of decimals (say 50). If $P$ is periodic and smooth, I usually know how to do this. But otherwise, I run into problems. Consider the following two specific examples:

1) $$S_k(a)=\sum_{n\ge1}\dfrac{J_0(na)}{n^k}\;,$$ with $J_0$ a $J$-Bessel function. If $a$ is a rational multiple of $\pi$ the sum is very regular and I can compute it. Assume otherwise. When $k\ge1$, using the integral representation of $J_0$ I can compute the sum $S_k(a)$. But when $k=0$ I have no idea, although I believe that the series converges: so question 1:

Does the series $\sum_{n\ge0}J_0(n)$ converge, and if yes, how to compute 50 decimals of its sum ?

2) $$T_k(a)=\sum_{n\ge1}\dfrac{\{na\}-1/2}{n^k}\;.$$ This series was considered by Hecke. How to compute it numerically when $a$ is irrational ? Since Hecke gives some formulas when $a$ is quadratic, specific question 2:

is it possible to compute numerically to 50 decimals $\sum_{n\ge1}\{na\}/n^2$ for $a=\pi$, and if not, for $a=\sqrt{2}$ for instance ?