I am self studying Analytic number theory from Tom M Apostol Modular Functions and Dirichlet Series in Number theory and I have a doubt in calculating residue in Chapter -5 , page 95 .
The partition function p(n) generated by Euler 's Infinite Product
Define F(x) = $ \sum_{n=0} ^\infty p(n) x^n $ , p(0) = 1 .
Here x[n+1] means x is being raised to power n+1 .
If 0<|x|<1 , then dividing both sides by $ x^{n+1} $ $\frac {F(x) } { x^{n+1}} $ = $\sum_{k=0} ^\infty \frac { p(k) x^k } { x^{n+}} $ for each n greater than or equal to 0.
I understand that last series is Laurent expansion of $ \frac { F(x) } { x^{n+1}} $ in the punctured disk 0<|x|<1 .
Can somebody please give a proof of this
This function has a pole at x=0 with residue p(n) .
I am not able to prove how this residue is p(n) . Can somebody please help.