It converges.
You may perform the straightforward integration by parts way, but I'll perform three ways to tackle this.
Solution #1:
Substitute
Solution #2:
Construct a double integral to reverse integration order, the rest follows easily.
Solution #3:
By the geometric series
This yields Hence
The last sum is a telescoping series.
Thanks for that useful thumbnail, Jhevon. I'll just add the following:
has the indeterminant form . The re-write to puts it into a 'standard' indeterminant form.
Note that ' '
(and you can count how many mathematical cows I've slaughtered here. But that's OK - slaughtering mathematical cows is something physicists do best )