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Math Help - squeeze theorem

  1. #1
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    squeeze theorem

    cos(n*pi/n^2)

    i have to prove that this sequence is converging to 0. i am unable to see how this would converge to 0. any ideas?
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  2. #2
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    Quote Originally Posted by needhelp101 View Post
    cos(n*pi/n^2)

    i have to prove that this sequence is converging to 0. i am unable to see how this would converge to 0. any ideas?
    It doesn't converege to zero.

    What you have simplifies to \cos \frac{\pi}{n} and this converges to 1 as n --> +oo.
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  3. #3
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    yea, i was able to understand that, but according to the question, it is suppose to converge to 0. i was able to get it to converge to 1, but i wasn't sure if there was any other way for it to converge to 0.
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by needhelp101 View Post
    cos(n*pi/n^2)

    i have to prove that this sequence is converging to 0. i am unable to see how this would converge to 0. any ideas?
    As you have it written \lim_{n\to\infty}\cos\left(\frac{n\pi}{n^2}\right) = 1 because as n\to\infty, \frac{n\pi}{n^2}\to 0 and \cos(0)=1.

    Perhaps you meant \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2}? This does equal zero because \cos(n\pi)\leq 1 for all n so \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2} \leq \lim_{n\to\infty}\frac{1}{n^2} = 0
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  5. #5
    Junior Member woof's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    Perhaps you meant \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2}? This does equal zero because \cos(n\pi)\leq 1 for all n so \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2} \leq \lim_{n\to\infty}\frac{1}{n^2} = 0
    And to add to that, now the "squeeze" part:

    -1\leq\cos(n\pi)\leq 1 for all n so \lim_{n\to\infty}\frac{-1\ \ }{n^2} \leq\lim_{n\to\infty}\frac{\cos(n\pi)}{n^2} \leq \lim_{n\to\infty}\frac{1}{n^2}

    Now both "ends" converge to 0, squeezing the middle to zero.
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  6. #6
    Super Member redsoxfan325's Avatar
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    Right, I forgot to finish squeezing in my reply.
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  7. #7
    Junior Member woof's Avatar
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    But you got the important part
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  8. #8
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    thank u all so much. u all have been very helpful 2 me for this problem.
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