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Math Help - Evaluate the limit?

  1. #1
    Super Member fardeen_gen's Avatar
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    Evaluate the limit?

    Evaluate:
    \lim_{n\rightarrow \infty} \frac{1}{2^n}\left\{(1 + \cos x)(1 + \cos \frac{x}{2})(1 + \cos \frac{x}{4})\mbox{.....}(1+ \cos \frac{x}{2^{n - 1}})\right\}
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  2. #2
    MHF Contributor

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    Quote Originally Posted by fardeen_gen View Post
    Evaluate:
    \lim_{n\rightarrow \infty} \frac{1}{2^n}\left\{(1 + \cos x)(1 + \cos \frac{x}{2})(1 + \cos \frac{x}{4})\mbox{.....}(1+ \cos \frac{x}{2^{n - 1}})\right\}
    if x=0, the limit is clearly 1. for x \neq 0 we have f_n(x)=\frac{1}{2^n}\prod_{j=0}^{n-1} \left[1 + \cos \left(\frac{x}{2^j} \right) \right]=\prod_{j=1}^n \cos^2 \left(\frac{x}{2^j} \right)=\frac{\sin^2 x}{4^n \sin^2 (\frac{x}{2^n})} and thus \lim_{n\to\infty} f_n(x)=\frac{\sin^2 x}{x^2}.
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