Rational or irrational sum and the integral

Question

I wanted to ask you is it possible to define that the number n is rational or irrational from analysis of integral form of function of series, for e. x. we have series $$\sum_{n=1}^{\infty}\frac{1}{{n^2}}$$ and we don't know that the sum is rational or irrational, (we assume that we don't know that is $\frac{π ^2}{6}$). But we can calculate the integral

$${\int_{0}^{\infty}\frac{1}{n^2}\,dn=1}$$

Can we say something about sum, if it is rational or irrational without calculating it?

Answer 1

Short answer: no. The integral proves the sum converges by providing a bound on the values of the increasing sequence of partial sums. The fact that the bound is rational doesn't help trying to decide whether the sum is rational.

Answer 2

According to Wikipedia (which I deem trustworthy in this case), we can write the Euler-Mascheroni constant $\gamma$ as $$ \gamma=\sum_{n=1}^{\infty}\frac{|G_n|}{n} $$ where $G_n$ is the $n$th Gregory coefficient. The terms of the series are rational, but it's still unknown whether $\gamma$ is rational or irrational.

Another series expansion is $$ \gamma=\sum_{n=1}^\infty\left(\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right) $$ We could consider the integral $$ \int_1^\infty\left(\frac{1}{x}-\log\left(1+\frac{1}{x}\right)\right)\,dx=2\log2-1 $$ (if my computation is correct). This is irrational, actually transcendental, but cannot give insight on the nature of $\gamma$.