I'm doing complex analysis and the section on convergence of series to infinity has me stumped 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.