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Prove roots lie inside the unit circle

  • Thread starter rioo
  • Start date
  • #1
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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?
 

Answers and Replies

  • #2
Have you had Rouches theorem in class yet?
 

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