## Summing the first $n$ powers and generating functions

One of the classic and most used sums in all of mathematics is the sum of the first $n$ natural numbers $$1 + 2 + \cdots + n = \frac{n(n+1)}{2}.$$ Gauss's classic derivation of this formula involves observing that if we duplicate this sum, write it backwards under the first sum, and add termwise, we […]