Final answer:
To find the convergence and radius of convergence of the given series 1/x^n x^(-n), we can use the ratio test. The series converges if |x| < 1, and the radius of convergence is 1.
Step-by-step explanation:
To find the convergence and radius of convergence of the given series, which is \sum\limits_{n=1}^{\infty}\frac{1}{x^n}x^{-n}, we can use the ratio test. The ratio test states that if L = \lim\limits_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| exists, then the series converges if L < 1 and diverges if L > 1. In this case, we have a_n = \frac{1}{x^n}x^{-n}.
By simplifying this expression, we get a_n = \frac{1}{x^n}. Applying the ratio test, we have L = \lim\limits_{n\to\infty}\left|\frac{\frac{1}{x^{n+1}}}{\frac{1}{x^n}}\right| = \lim\limits_{n\to\infty}|x| = |x|. So, the series converges if |x| < 1 and diverges if |x| > 1. The radius of convergence can be found by solving |x| = 1. Therefore, the radius of convergence is 1.