$\displaystyle {prove \ that \ if \ \mid x_{n+1} - x_n \mid \leq \frac{1}{3^n} , \forall n \in N ,\ then \ x_n \ is \ a \ cauchy \ sequence\ .}$
Just as you said, $\displaystyle |x_J - x_K| \le \sum_{k = K}^{J-1} \frac{1}{3^k} $.
That's a finite sum, and so for large enough k, the entire sum should go to 0. But if we're trying to prove that $\displaystyle |x_J - x_K| < \epsilon $ for all J, K > k, then isn't it possible to hold K fixed and take J so large that the sum doesn't actually approach 0?
Let me know if my question isn't clear enough.