Consider the series
$$ \sum^\infty_{n=1} \frac{1}{n(n+1)(\log(n+2))} $$
Find a partial sum $S_n$ that approximates "$S$" with an error smaller than $0.1,$ and deduce if $S_n$ is greater than or smaller than $S.$
How am I supposed to know till which $n$ I am supposed to find the sum. Any help will be fine.
Thank You
You are supposed to find an $N$ such that $$\sum^{\infty}_{n=N}\frac{1}{n(n+1)(\log(n+2))}\lt 0.1$$ Note that they did not ask for the minimum $N$, just one that is large enough to guarantee the inequality. Since $\log(n+2) \gt 1$ we can say $$\sum^{\infty}_{n=N}\frac{1}{n(n+1)(\log(n+2))} \lt \sum^{\infty}_{n=N}\frac{1}{n(n+1)}$$ The last sum telescopes, so finding an $N$ that makes it less than $0.1$ is not hard. For the second part, all the terms are positive, so the partial sums are always less than the limit.