1. ## Complex convergence

Hey,

At first I thought part a and c both converged but now after reading that "A sequence converges precisely when both the Im(z) and the Re(z) converge" which in this set of questions makes each of them converge. Is that correct?

Part b seems to repeat after 6ish n's and continues to repeat for large n which makes me think it doesn't converge.

Also would the limit for a and c be zero? as cos/sin are between -1 and 1 and (c) because sqrt(3)^n > sqrt(2) which is the norm of (1+i)

Daniel

2. ## Re: Complex convergence

(a) For $\displaystyle z_n=\frac{\cos n\theta+i\sin n\theta}{n}$ , real and imaginary parts converge to $\displaystyle 0$ so, ...

(b) For $\displaystyle z_n=\frac{(1+i)^n}{(\sqrt{2})^n}=\frac{(\sqrt{2})^ n(\cos (n\pi/4)+i\sin (n\pi/4))}{(\sqrt{2})^n}=\cos (n\pi/4)+i\sin (n\pi/4)$ so ...

(c) For $\displaystyle z_n=\frac{(1+i)^n}{(\sqrt{3})^n}$ , $\displaystyle |z_n|\to 0$ so ...

3. ## Re: Complex convergence

Thanks Fernando,

So just to double check,

I was right to think b diverges because e^(i n pi/4) ->inifnity as n -> infinity

4. ## Re: Complex convergence

Originally Posted by Daniiel
I was right to think b diverges because e^(i n pi/4) ->inifnity as n -> infinity
No, it is an oscillating sequence.