<s>[INST] Prove that $\lvert\frac{1}{n}-\frac{1}{2n}-\frac{1}{2n+2}\rvert<\frac{1}{n^2}$ Prove that $\lvert\frac{1}{n}-\frac{1}{2n}-\frac{1}{2n+2}\rvert<\frac{1}{n^2}$ and deduce that $1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}$ is convergent. 
Using algebra, the absolute value becomes $\lvert\frac{1}{2n}-\frac{1}{2n-2}\rvert$ which is $\lvert\frac{-(n+1)}{2n(n+1)}\rvert$.
Not entirely sure how to proceed with this proof..
Edit: Incorrect third term, changed to $\frac{1}{2n+2}$.
 [/INST] $$
\begin{array}{rcl}
\left| \frac{1}{n}-\frac{1}{2n}-\frac{1}{2n-2} \right| & < & \frac{1}{n^2} \\
\left| \frac{1}{2n}-\frac{1}{2n-2} \right|& < & \frac{1}{n^2} \\
\left| \frac{-2}{(2n)(2n-2)} \right|& < & \frac{1}{n^2} \\
\frac{1}{2n^2-2n}& < & \frac{1}{2n^2-n^2} \\
\end{array}
$$
True for all integer $n>1$.
</s>