I need a little help in summing the the following series:
$ 1+2v^4+3v^8+4v^{12}+\ldots + 20v^{76}?$
Is there a closed formula for summing $ \sum_{k=0}^{n} k\cdot ar^k?$
I need a little help in summing the the following series:
$ 1+2v^4+3v^8+4v^{12}+\ldots + 20v^{76}?$
Is there a closed formula for summing $ \sum_{k=0}^{n} k\cdot ar^k?$
\begin{align} S & = \sum_{k=1}^n akr^k = ar \sum_{k=1}^{n} k r^{k-1} = ar \dfrac{d}{dr} \left( \sum_{k=1}^{n} r^{k}\right)\\ & = ar \dfrac{d}{dr} \left( \dfrac{r(r^n-1)}{r-1}\right) = ar \left(\dfrac{nr^{n+1} - (n+1)r^n + 1}{(r-1)^2} \right) \end{align}