Determine if sum of series is rational or not

Solution 1:

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

Solution 2:

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.

Solution 3:

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.