Geometric series

Derived sums

We notice that \[\frac{d S_n}{dr} = \sum \limits_{k=0}^{n} a k r^{k-1}=a \frac{(n+1)r^n(r-1)- r^{n+1}+1}{(r-1)^2}\]

The last part comes from the sum formula. It simplifies (somewhat) to: \[\sum \limits_{k=0}^{n} a k r^{k-1}=a \frac{n r^{n+1}-(n+1)r^n +1}{(r-1)^2}\]

Higher derivatives allow the determination of sums of the form \(\sum \limits_{k=0}^{n} a k^p r^{k-1}\).