Application of Cauchy's Double Series Theorem

[Ed Quanta]

I have to use Cauchy's Double Series Theorem and the following equation,

1/(1-z)^2= 1 + 2z + 3z^2 + 4z^3 + 5z^4+...

to prove that

z/(1+z) - 2z^2/(1 + z^2) + 3z^3/(1+z^3)-+...=
z/(1+z)^2 - z^2/(1+z^2)^2 + z^3/(1+z^3)^2-+...

Any hints on how to start?

Note, |z|<1

(I am not sure, but I think it might be easiest to prove this true where z is real, and then use the identity theorem to show this is true where z is complex)

 

[Hurkyl]

Double series theorem? You mean interchanging summation signs?

Anyways, have you tried writing these two expressions as infinite sums?

 

[Ed Quanta]

Double series theorem says a series amn is convergent if and only if |amn|< infinity in which case both iterated sums are equal (In other words if you sum with respect to n first and then m, or vice versa) in which case both iterated sums converge.