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Math Help - help with a limit

  1. #1
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    help with a limit

    I'm pretty sure this limit is zero, but I was hoping someone could help me show it with algebra:


    Suppose that lim as n goes to infinity of (an) = 0. Let (bn) be a non negative sequence with lim as n goes to infinity of (b1 + b2 + + bn1 + bn) = infinity.

    What is the limit as n goes to infinity of:
    (a1b1 + a2b2 + + an1bn1 + anbn)
    ------------------------------------- ?
    (b1 + b2 + + bn1 + bn)



    Should you divide everything by anbn or just bn? I'm not sure.



    Thank you for any help.
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  2. #2
    MHF Contributor red_dog's Avatar
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    We'll apply the Stolz-Cesaro lemma:
    If (x_n) and (y_n) are two sequences such that (y_n) is ascending and unbounded and if exists \displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=l, then \displaystyle\lim_{n\to\infty}\frac{x_n}{y_n}=l.

    Now, let x_n=a_1b_1+a_2b_2+\ldots+a_nb_n, \ y_n=b_1+b_2+\ldots+b_n

    We have y_n<y_{n+1} and \lim_{n\to\infty}y_n=\infty, so (y_n) is unbounded.

    \displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=\lim_{n\to\infty}\frac{a_{n+1}b_{n+1}}{b_{n+1  }}=\lim_{n\to\infty}a_{n+1}=0.
    So, \lim_{n\to\infty}\frac{x_n}{y_n}=0
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  3. #3
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    I really appreciate the help. Is there a more complicated (or less direct) way to do it because I can't use that theorem in my class. It makes sense, but I looked at the proof of it and it's kind of out of my league.

    Would you be able to prove the Stolz-Cesaro theorem with just elementary real analysis?
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  4. #4
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    Don't worry about it. Thanks for the help.
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  5. #5
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    Quote Originally Posted by red_dog View Post
    We'll apply the Stolz-Cesaro lemma:
    Can you teach me this lemma by posting it in this here.

    I was going to add more tricks but I keep forgetting.
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