Closed form for the sum?

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.

• I don't see a reason to relate this sum to Zeta. – Yves Daoust May 9 at 21:43
• I can't answer such a negative question. Why do you see a relation ? – Yves Daoust May 9 at 21:46
• This doesn't mean that they are related. You could as well see terms of geometric sequences or some Taylor development. – Yves Daoust May 9 at 21:52
• Said differently, picking terms from different series gives an unrelated series, because a single term tells nothing about a series. You can't retrieve $\zeta(4)$ from $1/3^4$. – Yves Daoust May 9 at 21:57
• Just for fun though, this may be relatable to the Sophomore's dream – Clement C. May 9 at 21:59