You've established that
so all you need to do is compute the integral. Rewrite the integrand as
and replace
, so that 
. Then
By the fundamental theorem of calculus, this evaluates to
and so the series converges.
You also could have used the fact that the series is geometric with common ratio less than 1 to arrive at the same conclusion, with the added perk of being able to find the exact value of the sum to corroborate this.
so all you need to do is compute the integral. Rewrite the integrand as
and replace
By the fundamental theorem of calculus, this evaluates to
and so the series converges.
You also could have used the fact that the series is geometric with common ratio less than 1 to arrive at the same conclusion, with the added perk of being able to find the exact value of the sum to corroborate this.