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Math Help - Complex sequence

  1. #1
    Newbie Hugal's Avatar
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    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 :

    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.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Hugal View Post
    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
    Last edited by chisigma; October 12th 2010 at 07:07 PM.
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  3. #3
    Newbie Hugal's Avatar
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    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.
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