Power series converges

Question

The below power series converges at x = 5. For which of the following values of x must the series converge?

+infinity

∑ a_n(x-1)ⁿ

n=1

a. x=-3

b. x=-1

c. x=7

d. x=-5

Answer 1

The power series is centered at x = 1, so the interval of convergence has center x = 1 and converges at x = 1

Since the series is known to converge when x = 5, which is 4 units to the right of 1, then the series converges for all values of x in the interval [1,5] and, by symmetry, it must also converge at x = -3, which is 4units to the left of 1, and for all values of x in the interval [-3,1]. So, the interval [-3, 5] is at least part of the interval of convergence.

Answer: a (-3) and b (-1)