Is there a closed form solution for the sum of an infinite geometric series, where the growth from element n to element n+1 converges to zero for $n\rightarrow\infty$?
Specifically, I'm looking for a closed formula for $$\sum_{t=0}^\infty\left(1+a\phi^t\right)^t\omega^t$$ where $\omega<\phi<1$ as well as $a<1$.
Alternatively, a solution for $$\sum_{t=0}^\infty\left(1+\frac at\right)^t\omega^t$$ would also be great.
Thanks a lot for looking into this.