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Math Help - Prove a sequence {x_n} converges

  1. #1
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    Prove a sequence {x_n} converges

    Let x_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} - \ln(n) where n = 1,2,3,...


    Prove that the sequence {x_n} converges.


    Hint: Show, first, that x > \ln(1+x) for any x > 0. Then show that the sequence {x_n} increases and x_{n+1} - x_n < \frac{1}{2(n+1)^2}. Use the theorem about the convergence and divergence of p-series to complete the proof.

    How should I proceed?
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  2. #2
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    Re: Prove a sequence {x_n} converges

    Hey vidomagru.

    If you can show that the difference is < 1/(2(n+1))^2 then the ratio test should be sufficient to show convergence of the sequence.

    Think of a power series and how the ratio test is used to show convergence (in terms of an+1/an where an is the nth coefficient).

    The other hint: Use the x > ln(1+x) to bound the difference that takes into account the logarithm term (i.e. ln(1+x) - ln(x) < x+1-x = 1).
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    Re: Prove a sequence {x_n} converges

    Hi,
    Do you have a typo in your sequence? As written, the sequence xn is the defining sequence of Euler's constant gamma. Moreover, the given sequence is not increasing, but decreasing!
    Let H_n=\sum_{k=1}^n{1\over k}

    \gamma=\lim_{n\rightarrow\infty}H_n-\text{ln}(n)=\lim_{n\rightarrow\infty}x_n

    \gamma\approx.577
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    Re: Prove a sequence {x_n} converges

    Quote Originally Posted by chiro View Post
    Hey vidomagru.

    If you can show that the difference is < 1/(2(n+1))^2 then the ratio test should be sufficient to show convergence of the sequence.

    Think of a power series and how the ratio test is used to show convergence (in terms of an+1/an where an is the nth coefficient).

    The other hint: Use the x > ln(1+x) to bound the difference that takes into account the logarithm term (i.e. ln(1+x) - ln(x) < x+1-x = 1).
    Ok, I am going to try to figure this part out and will get back to you on that, but in the mean time it seems like the sequence is decreasing not increasing?
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    Re: Prove a sequence {x_n} converges

    Could I use the monotone convergence theorem here?
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    Re: Prove a sequence {x_n} converges

    Hi again,
    I guess this is really the sequence you want. Yes, the monotone convergence theorem applies. Here's a complete proof:

    Prove a sequence {x_n} converges-mhfcalc20.png
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