Q.Find $\displaystyle\sum_{r=1}^{\infty}\left(-\dfrac13\right)^{r-1}$.

For a G.P. with first term $a$ and common ratio $r$, the sum of the first $n$ terms is $S_n=\dfrac{a(1-r^n)}{1-r}$. If $|r|<1$, $r^n\to0$ as $n\to\infty$, so $S_n$ approaches a fixed finite value $\dfrac{a}{1-r}$, called the sum to infinity, $S_\infty=\dfrac{a}{1-r}$. If $|r|\ge1$, $r^n$ never settles down as $n$ grows, so the infinite sum has no finite value and does not exist. This single condition, $|r|<1$, is therefore the deciding test before attempting to sum an infinite G.P. -- testing it first avoids applying the formula to a series that does not actually c …