Final answer:
To find the range of x-values for which the ratio test implies that the series converges, analyze the power series and determine the values of x for which the limit of (xn+1)/xn is less than 1. The range of x-values for convergence is (-∞, ∞) or all real numbers.
Step-by-step explanation:
To determine the range of x-values for which the ratio test implies that the series converges, we need to analyze the power series. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. For the power series with term xn, the ratio of consecutive terms is given by (xn+1)/xn. Therefore, the range of x-values for convergence is obtained by finding the values of x for which the limit of (xn+1)/xn is less than 1.
For example, if the power series is ∑(xn/n!), then the ratio of consecutive terms is (xn+1/(n+1)!)/(xn/n!) = xn+1/(n+1)*xn. Taking the limit as n approaches infinity, we get |x/(n+1)|, which is less than 1 when |x| < infinity. Hence, the range of x-values for convergence is (-∞, ∞) or all real numbers.