Question

Find the radius of convergence, R, of the series X^(n) / 5n-1 Find the interval, I, of convergence of the series. (Enter your answer using interval notation.)

Answer 1

To determine the radius of convergence for the series X^(n) / (5n - 1), the Ratio Test is used, resulting in a radius of convergence R of 5.

Step-by-step explanation:

To find the radius of convergence and the interval of convergence for the given series X^(n) / (5n - 1), we can apply the Ratio Test. The Ratio Test tells us that if the limit of the absolute value of the ratio of consecutive terms of a series (as n approaches infinity) is less than 1, the series converges. Therefore, for our series, we take the limit as n approaches infinity of |(X^(n+1) / (5(n+1) - 1)) / (X^n / (5n - 1))| which simplifies to the limit of |X / 5|. The series converges when this limit is less than 1, which occurs for |X| < 5. Thus, the radius of convergence R is 5.