Basic Geometric Series Question-Stuck I'm studying Calc 2 and I have a basic series question.
A geometric series $\sum _{n=0}^{\infty }\:Ar^n$ is convergent if |r|<1 and the sum equals $\frac{a}{1-r}$ if the series is convergent.
Question: $\sum _{n=1}^{\infty }\:\frac{5}{\pi ^n}=-\frac{5}{\pi }+\sum _{n=0}^{\infty \:}\:\frac{5}{\pi ^n}$
and $\sum _{n=0}^{\infty \:}\:\frac{5}{\pi ^n}$ totals $\frac{5}{1-\frac{1}{\pi }}=\frac{5\pi \:}{\pi \:-1}$.
Therefore, $\sum _{n=1}^{\infty }\:\frac{5}{\pi ^n}=-\frac{5}{\pi }+\sum _{n=0}^{\infty \:}\:\frac{5}{\pi ^n}$ = $\frac{5\pi \:}{\pi \:-1}\:-\frac{5}{\pi }$.
I know this is wrong per my textbook, but after an hour working on this problem I cannot figure out my error.


I know this is wrong per my textbook, but after an hour working on this problem I cannot figure out my error.

$-\dfrac5{\pi^0}=-5$, because $\pi^0=1$.
