1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Complex analysis proof

  1. Jun 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Given |z|<1 and n a positive integer prove that
    \left|\frac{1-z^n}{1-z}\right|\le n

    3. The attempt at a solution
    I try to find the maximum of the function by differentiation

    \frac{d}{dz}\frac{1-z^n}{1-z}=\frac{-nz^{n-1}*(1-z)+(1-z^n)}{(1-z)^2}=0\Rightarrow (1-z^n)=nz^{n-1}*(1-z)

    I then plug this in

    \left|\frac{nz^{n-1}*(1-z)}{1-z}\right|=n\left|z^{n-1}\right|\le n

    I guess this works. Does someone have another way to prove it?
  2. jcsd
  3. Jun 15, 2009 #2


    User Avatar
    Homework Helper

    Divide the expression inside absolute values and see what you have - use the bound on |z|.
  4. Jun 15, 2009 #3
    Ah that is clever

    \left|\frac{1-z^n}{1-z}\right|=\left|\sum_{k=0}^{n-1}z^k\right|\le \sum_{k=0}^{n-1}|z|^k\le n
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook