# Math Help - Remainder Estimate Problem

1. ## 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.

2. ## Re: Remainder Estimate Problem

Originally Posted by Beevo
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