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Limit problem ( can't find the mistake)

  1. May 26, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi. My professor asked me if I know to solve this limit and I tried doing it, however I didn't get the same answer as him.
    Question: What is the limit of (cos(x)*cos(2x)*cos(3x)*...*cos(nx)-1)/x^2 as x approches 0
    20160526_131520.jpg
    2. Relevant equations
    /
    3. The attempt at a solution
    20160526_131531.jpg

    So to find this limit I used L'Hospital's rule.
    I derived the top and bottom function and I noticed that each part of the derivative can be calculated using sin(x)/x as x ->0 =1. Then I noticed that the whole thing equals -(1^2+2^2+3^2+.....n^2)/2
    However my professor said that the whole thing should equal -n(n+1)(2n+1)/2
    Could somebody please find the part where I made the mistake
     
  2. jcsd
  3. May 26, 2016 #2

    ehild

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    Your result is correct but not complete. The sum of the squares of the first n integer is n(n+1)(2n+1)/6
    Your teacher made a mistake.
    http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm
     
  4. May 26, 2016 #3
    Thank you. So the final answer -n(n+1)(2n+1)/12 would be correct?
     
  5. May 26, 2016 #4

    haruspex

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    Yes.
     
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