Q.Find the sum to infinity of the arithmetic-geometric series $1 + 2\left(\dfrac12\right) + 3\left(\dfrac12\right)^2 + 4\left(\dfrac12\right)^3 + \ldots$

[!TLDR] A.G.P. with $a=1,d=1,r=\tfrac12$; apply $S_\infty=\dfrac{a}{1-r}+\dfrac{dr …

Here $a=1,d=1,r=\tfrac12$, and $|r|<1$ so the infinite A.G.P. formula from Section 7 applies: $S_\infty=\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2}=\dfrac{1}{1/2}+\dfrac{1\c …

Identify $a,d,r$, confirm $|r|<1$, and substitute directly into $S_\infty=\ …

Using the finite-sum A.G.P. formula (with leftover $r^n$ terms) instead of the simpler infinite-sum formula is …