Sum[HarmonicNumber[k]/k^2, {k, 1, m}]

That is

$$\sum_{k = 1}^{m} \frac{H_k}{k^2}$$


I will attach a screenshot for I don't even know how to write it. And ys, I have tried to copu-paste it as LaTeX and as input command but it does not work.

enter image description here

Can someone help me in understanding what it does mean?


Let us denote

$$s_m=\sum_{k = 1}^{m} \frac{H_k}{k^2}.$$

The MA output means that $s_m$ fulfills the recursion relation

$$(m+2)^3 s_{m+2}=1+(m+2) \left(2 m^2+6 m+5\right) s_{m+1}-(m+1)^2 (m+2) s_{m}$$ with initial conditions $$ s_1=1,\quad s_2=\frac{11}{8}.$$

  • $\begingroup$ Now it's clear! Thank you so much! $\endgroup$
    – xyzt
    Feb 8 at 13:31

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