Question

how many terms of the series 1/(n*(ln(n)^8)) from n=2 to infinity would you have to add to find the sum to within 0.01

Answer 1

You can get an idea from the continuous function
f(x)=1/(x*(ln(x)^8))
From:
∫f(x)dx from k to ∞
=(1/7)ln(k)^7
which gives an idea of the error if the series is summed to the kth term.

From the result, and equating
(1/7)ln(k)^7 = 0.01
you should get the value of k for the required precision.

Answer 2

thanks that makes sense