I'm pretty sure I haven't made any mistakes in part 1), but I have absolutely no idea how to do part 2). I don't even know what the question is asking and I've never seen anything like it before. I don't see how the sum subtracted from the sum can be anything other than 0?
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In the second question, it should be find an $N$ such that $$\left|S-\sum_2^N \frac{1}{n\ln^2 n}\right|\lt 10^{-3}.$$ The tail (sum from $N+1$ on) is less than $$\int_N^\infty \frac{1}{x\ln^2 x}\,dx.$$ The above integral is $\frac{1}{\ln N}$, which is $\lt 10^{-3}$ if $N\gt e^{1000}$. A big number! We can do somewhat, but not much better. The convergence of our original sum is very very slow.
$\frac 1{n\ln^2n}$ is a decreasing function. From the Reimann integral:
$\sum_\limits{n=j+1}^{k} \frac 1{n\ln^2i}<\int_j^k \frac1{\ln^2n}dn <\sum_\limits{n=j}^{k-1}\frac1{n\ln^2n}$
find $k$ such that: $\sum_\limits{n=k+1}^{\infty}\frac1{n\ln^2n}<\int_{k}^{\infty} \frac1{n\ln^2n}dn <10^{-3}$