1. ## Complex sequence

Hi everyone,
This is my first post here some i'm little worried : will i do someting wrong ? We'll see x)
That's because i'm french, and even if my english is not very very bad, i'm likely to make sometimes some mistakes. So, forgive me about that :')
I'm not quite sure i post in the right forum too. Well, i don't even get how the American education system works (pre-university, university, and all that...)
Anyway, i choose this forum 'cause it seems to be pretty serious, and that's exactly what i was looking for as student in mathematics, a place where i can help people and where people can help me too

Anyway, i dared to begin this post, this is for a reason. I just can't get over an exercice our teacher gave us.
Well, the first part of this exercice was about a sequence which was defined by :

$\displaystyle u_n = \prod_{k=1}^{n} \cos \frac{\theta}{2^k}$

with $\displaystyle \theta \in ]-\pi ; \pi]$

I (if i am not wrong) had to study it and found that $\displaystyle \lim_{n \to + \infty} u_n = 1$

Well, after that, there's another question which looks like this :

Let $\displaystyle \forall x \geq 0, z_{n+1} = \frac{z_n+|z_n|}{2}$
Determine the limit of $\displaystyle \left ( z_n \right )_{n\geq 0 }$ which depends of $\displaystyle z_0$.

And, i just can't see how this question is related to the previous one (because they must be connected in some way !). I've already find that
$\displaystyle \forall n \geq 1, |z_{n+1}| \leq |z_n|$

I, of course, don't want the answer, but just some tips which could help me in some way

Hugal.

2. Originally Posted by Hugal
Hi everyone,
This is my first post here some i'm little worried : will i do someting wrong ? We'll see x)... that's because i'm french, and even if my english is not very very bad, i'm likely to make sometimes some mistakes. So, forgive me about that...

... the first part of this exercice was about a sequence which was defined by :

$\displaystyle u_n = \prod_{k=1}^{n} \cos \frac{\theta}{2^k}$

I (if i am not wrong) had to study it and found that $\displaystyle \lim_{n \to + \infty} u_n = 1$...
Bienvenue Monsieur!... regarding the first part of Your question, consulting the italian translation of Schaums Mathematical Handbook of Formulas and Tables, 1968, M.R. Spiegel , I found at the pag. 188 this 'infinite product'...

$\displaystyle \displaystyle \frac{\sin \theta}{\theta} = \cos \frac{\theta}{2}\ \cos \frac{\theta}{4}\ \cos \frac{\theta}{8}\ \cos \frac{\theta}{16}\ ...$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Yep ! Correct :')
Thanks to you i was able to find where i was wrong
And anyway, i think i've found the limit of $\displaystyle \left ( z_n \right )_{n \in \mathbb{N}}$
If i'm not wrong again, this is $\displaystyle |z_0|\frac{\sin(\arg z_0)}{\arg z_0}$