A surprising usage of derivatives to calculate certain sums of binomial coefficients | intermediate

Episode 127.

A surprising usage of derivatives to calculate certain sums of binomial coefficients | intermediate.
A surprising usage of derivatives to calculate certain sums of binomial coefficients | general mathematics | intermediate level.
The sums of binomial coefficients multiplied by the power of the sum index | general mathematics | intermediate level.

Branch of mathematics: general mathematics.
Difficulty level: intermediate.

We find the sums $\sum_{k=0}^{\infty}{k\binom{n}{k}}$, $\sum_{k=0}^{\infty}{k^2\binom{n}{k}}$, $\sum_{k=0}^{\infty}{k^3\binom{n}{k}}$, and show how to find the sum $\sum_{k=0}^{\infty}{k^s\binom{n}{k}}$ for any given $s$.

Mathematics.
#Mathematics

The same video on YouTube:
https://youtu.be/c3MnV6Uk8nc

The same video on Telegram:
https://t.me/mathematical_bunker/152