Demonstrate that the following sequence is a Cauchy sequence

Question

This is the sequence $$x_n = \sum _{k=1}^{n}\: \frac{e^{-k}}{k\left(k+2\right)}$$

And I need to demonstrate that it is a Cauchy sequence but I do not know how to proceed exactly, since it is fine at start but then I got stuck.

Answer 1

Note that\begin{align}\left|\frac{e^{-(n+1)}}{(n+1)(n+3)}+\cdots+\frac{e^{-(n+p)}}{(n+p)(n+p+2)}\right|&=\frac{e^{-(n+1)}}{(n+1)(n+3)}+\cdots+\frac{e^{-(n+p)}}{(n+p)(n+p+2)}\\&<\frac1{(n+1)(n+3)}+\cdots+\frac1{(n+p)(n+p+2)}\\&<\frac1{(n+1)^2}+\cdots+\frac1{(n+p)^2}\\&<\sum_{k=n+1}^\infty\frac1{k^2}\\&=\sum_{k=1}^\infty\frac1{k^2}-\sum_{k=1}^n\frac1{k^2}.\end{align}Now, given $\varepsilon>0$, take $N\in\Bbb N$ such that $\displaystyle\sum_{k=1}^\infty\frac1{k^2}-\sum_{k=1}^n\frac1{k^2}<\varepsilon$. Such a number $N$ must exist, since the series $\displaystyle\sum_{k=1}^\infty\frac1{k^2}$ converges.