Is there some methods to find out if the sum of an infinite series is rational or not if we have no closed form expression for the sum?
For instance:
$\begin{align*} &\sum_{n=1}^{\infty} \frac{n^2}{n!+1}\\ &\sum_{n=1}^{\infty} \frac{1}{n^{7/2} p_n}\\ &\sum_{n=1}^{\infty} \frac{n}{F_n}\end{align*}$
where $p_n$ is the $n$th prime number and $F_n$ is the $n$th Fibonacci number.
And if a sum of a series have no provable closed form expression, can it still be rational?
And is there a series which is known to be rational, but not which rational?