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Math Help - convergence of sequence via inequation

  1. #1
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    convergence of sequence via inequation

    Hi! I would like to proof that a_n:=\ln(n!)+n-(n+\frac{1}{2}) \ln(n) converges. I already have that
    (n+\frac{1}{2}) \ln(n+\frac{1}{2})-n \leq \ln(n!) \leq (n+1) \ln(n+1)-n or equivalently
    (n+\frac{1}{2}) \ln(1+\frac{1}{2n}) \leq \ln(n!)+n-(n+\frac{1}{2}) \ln(n) \leq (n+\frac{1}{2}) \ln(1+\frac{1}{n})+ \frac{1}{2} \ln(n+1)

    I have also calculated the difference a_n - a_{n+1}=(n+\frac{1}{2}) \ln(1+\frac{1}{n})-1 but i do not know how to use my inequations.

    Could somebody please help me with this?
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  2. #2
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    Quote Originally Posted by gammafunction View Post
    Hi! I would like to proof that a_n:=\ln(n!)+n-(n+\frac{1}{2}) \ln(n) converges. I already have that
    (n+\frac{1}{2}) \ln(n+\frac{1}{2})-n \leq \ln(n!) \leq (n+1) \ln(n+1)-n or equivalently
    (n+\frac{1}{2}) \ln(1+\frac{1}{2n}) \leq \ln(n!)+n-(n+\frac{1}{2}) \ln(n) \leq (n+\frac{1}{2}) \ln(1+\frac{1}{n})+ \frac{1}{2} \ln(n+1)

    I have also calculated the difference a_n - a_{n+1}=(n+\frac{1}{2}) \ln(1+\frac{1}{n})-1 but i do not know how to use my inequations.

    Could somebody please help me with this?
    I may be wrong, but I feel like your inequality is useless: the lower bound tends to 1/2, while the upper bound diverges to +\infty (because of the last log).

    However, if you know series, here's a way: find an asymptotic equivalence for a_{n+1}-a_n in order to show that \sum_n (a_{n+1}-a_n) converges. Try to see why this is enough.
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  3. #3
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    Thank you for the answer.
    Exactly that was my problem! In order to proof that a_n converges i was given the hint to use \ln(n!)=\ln(1)+...+\ln(n) and the inequations \int \limits_{k-\frac{1}{2}}^{k+\frac{1}{2}} \ln(x) dx \leq \ln(k) \leq \int \limits_k^{k+1} \ln(x) dx. Do you know how i could use this somehow?

    I do not really know how to work out your idea but i will think about it. Again i am thankful for every answer.
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  4. #4
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    Sorry, but i also do not get behind your idea... Can somebody please help me out? How do i use these estimations??
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