The series is converges if there exist a finite limit of partial sum sequence and is the sum of the series .
If the limit isn't exist, we say that the series is divergent.
Now, to telescopic series.
is telescopic series.
Partial sum of is:
By the definition above converges if exist.
Hence, , so converges iff converges.
In your case:
is converges sequence, so the series is also converges.
and , hence