Geometric series

Geometric series are sums of terms of the form \(z_n=a r^n\) with \(a,r\) constants.

Define \(S_n = \sum \limits_{k=0}^{n} a r^k\).

Multiplying by \(r\) and using the distributive law: \[r S_n=\sum\limits_{k=0}^{n}a r^{k+1}\]

We notice that this series includes all the terms of \(S_n\) except for the \(k=0\) term, \(a\). We also have an extra term, \(a r^{n+1}\): \[r S_n=S_n +a r^{n+1} -a\]

Solving and simplifying: \[S_n=a \frac{r^{n+1} -1}{r-1}\]