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Math Help - DIfferentiation Problem ^_^

  1. #1
    Junior Member mathbuoy's Avatar
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    Lightbulb DIfferentiation Problem ^_^

    Given f(n) = \frac{1}{4}n^2(n+1)^2. Evaluate \liminf\frac{f(n)}{f(2n)}.


    *show working and explanation PLS
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  2. #2
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    Quote Originally Posted by mathbuoy View Post
    Given f(n) = \frac{1}{4}n^2(n+1)^2. Evaluate \liminf\frac{f(n)}{f(2n)}.


    *show working and explanation PLS
    Hey mate,

    f(n) = (1/4)n^2(n+1)^2
    Thus,
    f(2n) = (1/4)(2n)^2(2n+1)^2 = n^2(2n+1)^2
    Hence,
    f(n)/f(2n) = (1/4)n^2(n+1)^2/ ( n^2 (2n+1)^2) = (1/4)(n+1)^2/(2n+1)^2

    to evaluate the limit of this expression as n tends towards infinity can be evaluated a number of ways, however given your title has the word differentation I am assuming your looking to use L'hopitals Rule,

    lim (n-->inf) g(n)/h(n) = lim(n-->inf) g'(n)/h'(n) if g(inf)/h(inf) = inf/inf

    Thus,

    lim (n-->inf) (1/4)(n+1)^2/(2n+1)^2 = (1/4)lim(n-->inf) ( 2(n+1)/(4(2n+1) )
    = (1/8)lim(n-->inf)( (n+1)/(2n+1) ) which once again uses inf/inf, thus employ L'hopitals again,

    =(1/8)lim(n-->inf) ((1)/(2)) = (1/8)lim(n-->inf)(1/2) = (1/8)(1/2) = (1/16)

    Hope this helps,

    David
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