Prove roots lie inside the unit circle

[rioo]

Homework Statement

Let P(z)=1+2z+3z^2+...nz^(n-1). By considering (1-z)P(z) show that all the zeros of P(z) are inside the unit disk

Homework Equations

None given..

The Attempt at a Solution

Well (1-z)P(z) = 1+z+z^2+...+nz^n
and to find roots I set it to 0:
1+z+z^2+...+nz^n = 0
This is a geometric series of z^n from z^0 to z^n-1 plus nz^n, so
(1-z^n)/(1-z) + nz^n = 0
1-(1-n)z^n-nz^(n+1) = 0

I have no idea where to go from here, we did nothing in class that gives me much idea where to go. we did some convergense stuff in class with the M test but that seems worthless here since i want roots, not convergence. am I on the right track with the geometric series or should I try something else?

 

[susskind_leon]

Have you had Rouches theorem in class yet?