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Math Help - Remainder Estimate Problem

  1. #1
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    Remainder Estimate Problem

    Hey guys,

    This problem asks: What is the smallest number of terms of the series from (1 to infinity) of 8/(n+1)(ln(n+1))^2 , you would need to add to find its sum to within 1/5.

    At first glance, I recognize that I have to find Rn<the integral of f(x) dx, so I used the integral with u-substitution and got 8/ln(n+1) < 1/5. Then by solving the inequality I got ln(n+1) > 40. Where would I go from here? Did I perhaps make a mistake in my calculations or do I have to simplify my solution.

    The answer choices are e^40, e^45, e^50, e^55, e^60.
    Any help and feedback is much appreciated, thanks.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Remainder Estimate Problem

    Quote Originally Posted by Beevo View Post
    Hey guys,

    This problem asks: What is the smallest number of terms of the series from (1 to infinity) of 8/(n+1)(ln(n+1))^2 , you would need to add to find its sum to within 1/5.....
    f[{n}]\text{:=}\sum _{k=1}^n \frac{8{Log}[k+1]^2}{k+1}




    f[n+1]-f[n] < 1/5 for all n>2429
    Attached Thumbnails Attached Thumbnails Remainder Estimate Problem-remainder-estimate-problem.png  
    Thanks from Beevo
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