- #1
daudaudaudau
- 302
- 0
Homework Statement
Given |z|<1 and n a positive integer prove that
[tex]
\left|\frac{1-z^n}{1-z}\right|\le n
[/tex]
The Attempt at a Solution
I try to find the maximum of the function by differentiation
[tex]
\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)
[/tex]
I then plug this in
[tex]
\left|\frac{nz^{n-1}*(1-z)}{1-z}\right|=n\left|z^{n-1}\right|\le n
[/tex]
I guess this works. Does someone have another way to prove it?