Answer
converges to $\dfrac{\pi}{\pi - e}$.
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
Here, $a=1$ and common ratio $r =\dfrac{e}{\pi}$
$S=\dfrac{a}{1-r}=\dfrac{1}{1-\dfrac{e}{\pi}}$
or, $S=\dfrac{\pi}{\pi - e}$
Hence, the series converges to $(\dfrac{\pi}{\pi - e})$