In undergraduate courses we compute the sum $S$ of some series of the form $\frac{1}{P(n)}$ where $P(x)$ is some simple polynomial of degree $2$ with integer coefficients, by the following procedure:
(sketch)
(a) Choose an appropriate periodic function $f(x)$ defined over a domain $D.$
(b) Compute the Fourier series $S(x)$ of $f(x).$
(c) Choose a suitable $x$ in $D$ so that we obtain a linear equation for $S.$
(d) Solve the equation to get $S.$
Example:
When $P(x)=x^2+1$ we can take:
$f(x)= \exp(x),$ $D= [-\pi,\pi[$, and $x=\pi.$
$S$ is the sum from $n=1$ to infinity of $\frac{1}{n^2+1}.$
We get the equation:
$$ ch(\pi) = S(\pi) = 2\frac{sh(\pi)}{2}(\frac{1}{2}+S2\frac{sh(\pi)}{\pi}(\frac{1}{2}+S) $$ that gives $$ S=\frac{1}{2}(\frac{\pi}{th(\pi)}-1). $$
($ch,sh,th$ denote the classic hyperbolic functions)
Question:
Why this fails (in general) for polynomials $P(x)$ of degree $3.$ ?
Why this fails for the polynomial $P(x)=x^3.$ ?
