# Limit of a complex sequence

## Homework Statement

find the limit z_n = [(1+i)/sqrt(3)]^n as n -> ∞.

## Homework Equations

3. The Attempt at a Solution

Apparently the limit is zero (via back of the book), but I have no clue how they got that answer.

(1 + i)^n seems to be unbounded, thus i do not see how z_n can go to zero I am lost.

Dick
Homework Helper

## Homework Statement

find the limit z_n = [(1+i)/sqrt(3)]^n as n -> ∞.

## Homework Equations

3. The Attempt at a Solution

Apparently the limit is zero (via back of the book), but I have no clue how they got that answer.

(1 + i)^n seems to be unbounded, thus i do not see how z_n can go to zero I am lost.

(1+i)^n is unbounded. But its absolute value is |1+i|^n. What's that??

Last edited:
(1+i)^n is unbounded. But its absolute value is |1+i|^n. What's that??

|1+i|^n = [sqrt(2)]^n?

Dick
Homework Helper
|1+i|^n = [sqrt(2)]^n?

Sure, so can you show the limit of |z_n| is 0? That would show the limit of z_n is also 0.

Sure, so can you show the limit of |z_n| is 0? That would show the limit of z_n is also 0.

that is not generally true,

take |(-1)^n + i/n| which converges to 1

{(-1)^n + i/n} does not converge.

jbunniii
Homework Helper
Gold Member
that is not generally true,

take |(-1)^n + i/n| which converges to 1

{(-1)^n + i/n} does not converge.
But it is true if the limit is zero. ##|z_n| \rightarrow 0## if and only if ##z_n \rightarrow 0##.

oh i see