TITLE: Determine if sum of series is rational or not
QUESTION [10 upvotes]: 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?

REPLY [12 votes]: There are some useful results obtained in these papers (and the references therein):

A theorem on irrationality of infinite series
and applications by C. Badea
The Irrationality of certain infinite series by C. Badea
A Theorem on Transcendence of Infinite Series II by M. A. Nyblom