Step by step
The question
Question: find for which x the series converges: the sum of xⁿ over n factorial.
Σn = 1∞xnn! =
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 to the n + 1, over x to the n, leaves one x.
The factorials leave n + 1.
|an+1an| = |x| × 1n + 1
2 The limit
1/n + 1 goes to 0.
So L is 0 for every x: always less than 1.
L = |x| × 0 = 0
3 The interval
The series converges for every x: the radius is infinite.
= all x: (−∞, ∞)
Careful!
L is 0 whatever x is, and 0 is less than 1.
So every x works.
Careful: L = 0 < 1 for every x
4 Check
Let's check an x inside, 3, with a calculator: 40 terms and 80 terms add up to almost the same, 19.0855.
It converges there.
Right!
x = 3: S40 ≈ 19.0855, S80 ≈ 19.0855
Let's look back.
First: the ratio.
Then: the limit.
Then: the interval.
Last: check.
Answer
= all x: (−∞, ∞)