Is there a closed form for this sum? It's a mixing summation of different terms in the zeta function with different values of $s.$
$$ S=\frac{1}{1^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{4^5}+ \cdot\cdot\cdot $$
Which can be written as:
$$ \zeta_1(2)+\zeta_2(3)+\zeta_3(4)+\zeta_4(5)+\cdot\cdot\cdot $$
where the $\zeta_1(2)$ denotes the first term of $\zeta(2).$
Or as:
$$ \sum_{i=1}^{\infty}(\frac{1}{i})^{i+1} $$
I don't think there is a closed form.