complex series absolute convergence?
I'm doing complex analysis and the section on convergence of series to infinity has me stumped (Headbang) so any help would be great.
The question is as follows:
Show that the series:
sum of [z/(z+1)]^n from n=0 to infinity
converges absolutely if and only if R(z)>-1/2 where R(z) is the real part of z.
To me this looks like a power series problem with the need to find the radius of convergence but I don't know what to do with the denominator. I tried substituting [z/(z+1)] in the expansion of sum of (z)^n from n=0 to infinity=1/(1-z) and then finding the remainder term pn but this is firstly just for convergence (not absolute) and secondly I got stuck.I even tried using the ratio test but once again I got stuck. I know that for a series to be absolutely convergent the series of the absolute value of [z/(z+1)]^n must be convergent where IzI=sqrt(x^2+y^2) for z=x+iy, but I don't know how to incorporate all these seperate peices of information into a meaningful answer.
Any help would be greatly appreciated.
Thanks in advance.