Friday, 30 January 2015

sequences and series - Proving suminftyn=1nzn=fracz(1z)2 for zin(1,1)





I do not know where to start, any hints are welcome.


Answer



Note that n=0zn=11z



for |z|<1.



Differentiating (1) and multiplying by z (this is legitimate since for any r<1, n=1nzn1 converges uniformly for |z|r<1) yields



zddzn=0zn=n=0nzn=n=1nzn=z(1z)2



for |z|<1.


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