Step by step
The question
Question: decide if the series converges: the sum of 5ⁿ over n factorial for n from 1 to infinity.
1 The next term
The ratio test: divide each term by the one before it, and let n grow.
If the limit L is less than 1, the terms shrink like a geometric series: it converges.
Bigger than 1: it diverges.
Exactly 1: the test cannot tell.
The next term: put n + 1 wherever there is an n.
2 The ratio
Dividing by aₙ is the same as times its flip.
Group the same kinds together.
5 to the n + 1, over 5 to the n, is just 5.
(n + 1) factorial is n + 1 times n factorial, so their fraction is 1/n + 1.
Careful!
(n + 1) factorial is n + 1 times n factorial.
So the fraction is n + 1, not 1.
3 The limit
As n grows: 1/n + 1 goes to 0.
So L = 0.
4 Decide
L = 0 is less than 1: the series converges.
5 Check
Let's check with a calculator: 10, 20, 50 terms add up to 145.3806, 147.4131, 147.4132.
It settles.
Right!
Let's look back.
First: the next term.
Then: the ratio.
Then: the limit.
Then: decide.
Last: check.