the limit does not hold, it's obviously false, can't you just see it?
as for the integral put and invoke the gamma function.
I have a sum .
Now, per partes it "is easy to show that" (for this I should use ).
I "know" that the holds ... but how to use it in per partes integration?
Let consider first the indefinite integral...
(1)
... with m and n non negative integers. Integrating by parts we obtain...
(2)
The integral in (2) is similar to the integral (1), but the exponent of log is n-1 instead of n. That suggests to integrate n times by part so that we obtain...
(3)
The second step is the computation of...
(4)
... and that is done applying n times l'Hopital's rule...
for (5)
Combining (3) and (5) with some easy steps we arrive to write...
(6)
... that for m=n=k becomes...
(7)
Kind regards