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I am trying to use Faulhaber's formula to determine partial sums of a power series.

Faulhaber's formula is given by

$\sum_{k=1}^{n}{k^{p}} = \frac{1}{p+1}\sum_{i=1}^{p+1}{(-1)^{\delta_ip}{p+1\choose i}}B_{p+1-i}n^{i}$ where $\delta_ip$ is the Kronecker delta function and $B_{p+1-i}$ is the $p+1-i$th Bernoulli number.

My question is, what do I use for $i$ in the Kronecker delta function when using this formula?

For example, I am trying to derive the partial sum of the power series $\sum_{k=1}^{n}{k^2}$ but I don't see what I would use for $i$.

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  • $\begingroup$ You mean $\sum\limits_{k=1}^n \color{red}k^2=\frac16\cdot n\cdot (n+1)\cdot (2n+1)$? $\endgroup$ – callculus Jan 5 at 18:20
  • $\begingroup$ Thank you @callculus. Yes, I will make that edit. $\endgroup$ – Gnumbertester Jan 5 at 18:21
  • $\begingroup$ $\delta_{ip} = 1 \space \mathrm{for} \space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$. $\endgroup$ – Andy Walls Jan 5 at 18:23
  • $\begingroup$ @AndyWalls , yes, I understand how to evaluate $\delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula. $\endgroup$ – Gnumbertester Jan 5 at 18:25
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    $\begingroup$ It is the index of the summation on the RHS, i.e. the index of each term in the summation. $\endgroup$ – Andy Walls Jan 5 at 18:40

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