MathDB

Complex number

Source: IMS 2006

July 11, 2006

limitfunctionalgebra proposedalgebra

Problem Statement

Suppose that

a_{1},a_{2},\dots,a_{k}\in\mathbb C

that for each

1\leq i\leq k

we know that

|a_{k}|=1

. Suppose that

\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.

Prove that

c=k

and

a_{i}=1

for each

i

.