Step by step
The question
Question: find for which x the series converges: the sum of n factorial (x − 1)ⁿ.
Σn = 1∞n!(x − 1)n =
1 The ratio
Use the ratio test.
x can be negative, so we take sizes, with bars.
ratio test: L = lim |an+1an| < 1
Divide the next term by this one.
(x − 1) to the n + 1, over (x − 1) to the n, leaves one (x − 1).
The factorials leave n + 1.
|an+1an| = |x − 1| × (n + 1)
2 The limit
n + 1 grows without end.
Unless (x − 1) is 0, L is infinity.
L = ∞ (if x ≠ 1)
3 The interval
The series converges only at x = 1: the radius is 0.
= only x = 1
Careful!
x = 1 still works: every term is 0 there.
Careful: not L = ∞, so no x works ✗
4 Check
Let's check: at x = 1 every term has a 0 in it, so the sum is 0: it converges there.
Right!
x = 1: every term is 0, the sum is 0
Let's look back.
First: the ratio.
Then: the limit.
Then: the interval.
Last: check.
Answer
= only x = 1