Nope, i'm not that good. I only know when i'm doing it incorrectly...
In general we want our summation technique to be regular, linear, and stable.
regular here loosely means that it sums convergent series correctly to the correct convergent value
linear means comes means you can add or subtract series from each other and the sums also add an subtract.
stable means adding a term at the beginnig of the series has the result of adding that value to the sum of the series.
In general once we start talking about a series like $1+2+3...=c$ we already leave the combo of linearity and stabilit…
A student recently asked whether there was a combinatorial proof of the following identity:
$\begin{equation*}
\sum^n_{k=1}(-1)^{n-k}k^2 = {n+1 \choose 2}.
\end{equation*}$
I was in a rush and couldn't come up with anything nice off the top of my head. Any ideas or pictures to make this clear...