What is the limit of the complex sequence z_n = [(1+i)/sqrt(3)]^n?

[DotKite]

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 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??

 

[DotKite]

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

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

 

[Dick]

|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.

 

[DotKite]

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]

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 $z_n \rightarrow 0$.

 

[DotKite]

oh i see