Since I'm new to analysis, I'm still never sure if my proofs are sound or have logical holes in them. Here's my proof for this one - hopefully you guys can find whatever logical errors are in the proof, or suggest a more efficient, slicker proof.
So, to show that a sequence is Cauchy, we look at the distance between consecutive terms further down the sequence. If we consider $x_{n}$ and $x_{m}$, then we see that if $m > n$ we can ignore all the terms in the difference of these two sums up to and including $\frac{n^2}{7^n}$ and consider the distance between $\frac{n^2}{7^n}$ and $\frac{(n+1)^2}{7^{n+1}}$, that is $\left|{x_{n} - x_{n+1}} \right|$, the distance $\left|{x_{n+1} - x_{n+2}} \right|$ and so on up to $\left|{x_{m-1} - x_{m}} \right|$. Now I'm too lazy to write it all down here, but we see that the distance between any two terms $\left|{x_{n} - x_{n+1}} \right|$ of this difference of sums is smaller than $(\frac{6}{7})(\frac{n^2}{7^n})$. So considering the sum of all these distances, we see that this is simply $(\frac{6}{7})(\frac{n^2}{7^{n}} + \frac{(n+1)^2}{7^{n+1}} +...+\frac{(m-1)^2}{7^{m-1}})$. The sum in the right bracket is less than one (or is it? why?...) so that whole expression is less than 6. Given $\epsilon > 0$, and with $N = 6$, $m, n > N$, and $ m \geq n$ we have that $\left|{x_{m} - x_{n}} \right| < \epsilon$, which shows the sequence is Cauchy.
Shoot..