Answer
Diverges
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
The series is a divergent geometric series with $a=1$ and common ratio $r =|\dfrac{e^\pi}{\pi^e}|$ or, $r==\dfrac{23141}{22.459}$
and $|r|=|\dfrac{23141}{22.459}| \gt 1$
Hence, the series diverges.