Step by step
The question
Question: decide if the series converges: the sum of n factorial over 3ⁿ 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.
3 to the n, over 3 to the n + 1, is 1/3.
(n + 1) factorial is n + 1 times n factorial, so their fraction is 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: n + 1 grows without end.
So L = infinity.
4 Decide
L = infinity is bigger than 1: the series diverges.
5 Check
Let's check with a calculator: 5, 10, 20 terms add up to 1.5679, 90.8957, huge.
It grows without end.
Right!
Let's look back.
First: the next term.
Then: the ratio.
Then: the limit.
Then: decide.
Last: check.