Complex sequence

**Hugal** · October 12th 2010, 10:44 AM

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 :

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

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

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

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

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

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
$\forall n \geq 1, |z_{n+1}| \leq |z_n|$

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

Thanks for taking your time to read me,

Hugal.

**chisigma** · October 12th 2010, 11:50 AM

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 :

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

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

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'...

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

Kind regards

$\chi$ $\sigma$

**Hugal** · October 13th 2010, 10:37 AM

Yep ! Correct :')
Thanks to you i was able to find where i was wrong

And anyway, i think i've found the limit of $\left ( z_n \right )_{n \in \mathbb{N}}$
If i'm not wrong again, this is $|z_0|\frac{\sin(\arg z_0)}{\arg z_0}$

Again, thanks for your fast answer

Hugal.

Thread: Complex sequence

LinkBack

Thread Tools

Display

Complex sequence

Similar Math Help Forum Discussions

Complex Sequence Limits

Randomized sequence of blocked 1-0-sequence and assignment of random integers

Complex Analysis, Sequence

Question about a complex sequence

set, geometric sequence,fraction and arithmetic sequence

Search Tags