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Math Help - Convergence

  1. #1
    Member iPod's Avatar
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    Convergence

    I have attached the question.

    I have expanded (1+h)^n using the binomial theorem, and the question after it I assume they want me to use the very same theorem to prove the convergence.

    So this is what I had done;

    (n+1)^{1/n}=1+h \rightarrow n+1=(1+h)^n=1+nh+\frac{n!}{2!(n-2)!}+...+h^n

    I'm not sure how to prove convergence from then on, but I am guessing the Sandwich Rule would need to be used here at one point.
    Any hints/tips?
    Attached Thumbnails Attached Thumbnails Convergence-binomial.png  
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  2. #2
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    Re: Convergence

    Let h_n be such that (n+1)^{1/n}=1+h_n. From what you wrote, 1+nh_n+n(n-1)/2\cdot h_n^2<1+n, so...
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  3. #3
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    Re: Convergence

    I understand how you derived the above inequality, however I'm not sure where to go from there - I'm not sure how you apply it to derive any convergence.
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  4. #4
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    Re: Convergence

    Can you show that h_n<f(n) for some decreasing function f? You don't have to solve the quadratic inequality nh_n+n(n-1)/2\cdot h_n^2<n for h_n; just find some function f.
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