Find the interval of convergence of the power series: $$\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 ) ^ { k }$$
A) (-1, 1)
B) Converges only when x = 1
C) (0, 1)
D) (-?, ?)

Question

Quizplus · Accepted Answer

The series is a power series centered at x = 1. To find the interval of convergence, we can use the ratio test. However, in this case, it's clear that the series diverges for all x except x = 1 due to the factorial term, which grows faster than any power of (x - 1) as k increases, unless x - 1 = 0 (which makes the series converge trivially to 0 for all terms after the first). Therefore, the series converges only when x = 1.

Find the Interval of Convergence of the Power Series ∑k=0∞k!(x−1)k\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 ) ^ { k }k=0∑∞​k!(x−1)k

Find the Interval of Convergence of the Power Series $\sum _ { k = 0 } ^ { \infty } k ! ( x - 1 ) ^ { k }$