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.

  • 7
    $\begingroup$ I don't see a reason to relate this sum to Zeta. $\endgroup$ – Yves Daoust May 9 at 21:43
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    $\begingroup$ I can't answer such a negative question. Why do you see a relation ? $\endgroup$ – Yves Daoust May 9 at 21:46
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    $\begingroup$ This doesn't mean that they are related. You could as well see terms of geometric sequences or some Taylor development. $\endgroup$ – Yves Daoust May 9 at 21:52
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    $\begingroup$ 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 $1/3^4$. $\endgroup$ – Yves Daoust May 9 at 21:57
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    $\begingroup$ Just for fun though, this may be relatable to the Sophomore's dream $\endgroup$ – Clement C. May 9 at 21:59

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