Ytukyg

This is Hewitt and Stromberg's Real and Abstract Analysis Problem 21.22. My approach was to compare with this question

How to derive $int_0^1 int_0^1 frac{1}{1-xy} ,dy,dx = sum_{n=1}^{infty}frac{1}{n^{2}}$

and use the series expansion of $frac{1}{(1-xy)^ p}$ to calculate this integral. I found this expansion to be

$$frac{1}{(1-xy)^ p}=sum_{n=0}^ infty frac{p cdots (p+n-1)}{n!} (xy)^ n.$$

However, I got this integral is

$$lim_{r to 1} int_0^ r int_0^ r (sum_{n=0}^ infty frac{p cdots (p+n-1)}{n!} (xy)^ n) dxdy = sum_{n=0}^ infty frac{pcdots (p+n-1)}{n!(n+1)^ 2}.$$

But, I don't seem to find if this series converges or diverges, much less what is the value of the integral. Also it says to calculate the following integrals,

$$int_{0}^ 1 int_{0}^ 1 frac{1}{(1-xy)^ p} dydx, int_{0}^ 1 int_{0}^ 1 Bigg|frac{1}{(1-xy)^ p}Bigg| dxdy, int_{0}^ 1 int_{0}^ 1 Bigg|frac{1}{(1-xy)^ p}Bigg| dydx$$

and to compare with Fubini's Theorem.

Have I done something wrong?

Thanks a lot!

This is a well-known integral for $0<p<1$ and involves the Harmonic number:

$$frac{H_{1-p}}{1-p}$$