Determine if sum of series is rational or not 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? 
 A: No, there are no such methods (if you're talking about a proof) except for some rather special cases.  Even with a closed-form expression, it is quite rare to be able to prove that something is irrational.  On the other hand, given a good numerical approximation to a number you can use continued fractions to see if this number appears to be a rational with small numerator and denominators.  For example, $\sum\limits_{n=1}^\infty \frac{n^2}{n!+1} \approx 4.0271515294669515849240298741047887364140370824913$ which has the continued fraction representation $\begin{split}4; & 36, 1, 4, 1, 8, 2, 3, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 2, 2, 6, 2, 33, 2, 1, 1, 1, 4, 18, 4, 1, 2, 6, 8, 3,\\& 1, 6, 1, 3, 1, 4, 4, 1, 9, 3, 8, 1, 2, 35 \ldots\end{split}$. This shows no signs of terminating, so the number is likely irrational.
A: It's unlikely that there are general methods. Witness the irrationality of $\zeta(2)$, which has a closed form $\pi^2/6$ found by Euler in 1735 (see Basel problem), but which was proved irrational by Hermite in 1873 only.  Witness also $\zeta(3)$, whose irrationality was proved only in 1978 by Apéry, but for which no closed form is known.
I guess the closest thing to a general method is Dirichlet's irrationality criterion and its generalizations such as Hurwitz's theorem. But even these are hard to apply in any given particular case.
A: 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
