Infinite Arithmetico-Geometric Series

Infinite Arithmetico-Geometric Series. The sum of the infinite AGP series $\sum\big((a+(n-1)d)\big)r^{n-1}$, for $-1<r<1$, is

This is simply the finite-AGP-sum formula (§5.5.1) with $n\to\infty$: as $-1<r<1$, both $r^n\to0$ and $r^{n-1}\to0$, so the finite-sum expression's $r^n$/$r^{n-1}$ terms vanish, leaving exactly the two terms above.

Example 5.19. For $1+\tfrac45+\tfrac7{25}+\tfrac{10}{125}+\cdots$ (here $a=1,d=3,r=\tfrac15$), …