# Understanding Kronecker Delta Function for Faulhaber's Formula

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 $$i$$.

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