Hey guys, I am currently in a DiffEq class, and I can't remember how to do Geometric Series for the life of me. I appreciate any help you can give me.
∑[(e^(-(i+1)πs)+e^(-iπs))/(s^2+1)]
The summation is from 0 to infinity.
Your series is $\displaystyle \frac{1}{1+s^2} \sum_{i=0}^\infty e^{-(i+1) \pi s} + e^{- i \pi s}$ which can be written as
$\displaystyle
\frac{1 + e^{\pi s}}{1+ s^2} \sum_{i=0}^\infty e^{- i \pi s} =
\frac{1 + e^{\pi s}}{1+ s^2} \sum_{i=0}^\infty \left(e^{- \pi s} \right)^i
$ = $\displaystyle
\frac{1 + e^{\pi s}}{1+ s^2} \cdot \frac{1}{1 - e^{-\pi s}}
$ provided that $\displaystyle e^{-\pi s} < 1$.
In general for infinite geometric series $\displaystyle S = a + ar + ar^2 + ar^3 + \cdots = \frac{a}{1-r}$ provided that $\displaystyle |r| < 1.$