Final answer:
To determine the values of x for which the series converges, we can use the ratio test. The convergence radius, sum of the series, convergence interval, and integral test are also explained.
Step-by-step explanation:
To determine the values of x for which the series converges, we can use the ratio test. The ratio test states that for a power series ∑ an(x - c)n, if the limit of |an+1(x - c)| / |an(x - c)| as n approaches infinity is less than 1, the series converges. If it is greater than 1, the series diverges. If it is exactly 1, the test is inconclusive.
In this case, the power series is centered at 1. So, we substitute c = 1. Then, we calculate the limit of |an+1(x - 1)| / |an(x - 1)| as n approaches infinity. Once we find the limit, we compare it to 1 to determine the convergence of the series.
The convergence radius (R) of the power series can be found by taking the reciprocal of the limit we obtained. The sum of the series can be calculated using the formula for the sum of an infinite geometric series. The convergence interval can be determined by checking when the series converges at the boundaries of the interval. Lastly, the integral test can be applied to further verify the results if needed.