Take the 2-minute tour ×  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?
share|improve this question I don't think there's even a known
closed form for 1. It seems likely that 3. is irrational based
on this, but I can't think of a proof. –  Ｊ. Ｍ. Sep 22 '11 at
0:49 I doubt we know any general method. But I guess a rule of
thumb is that the faster the denominators grow, the less likely
it is to be rational or algebraic (I base this guess on
Hurwitz's and Liouville's theorems). –  Joel Cohen Sep 22 '11 at
1:17 @Jo