Step by step
The question
Question: decide if the series converges: the sum of n² over 2ⁿ 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.
2 to the n, over 2 to the n + 1, is 1/2.
3 The limit
As n grows: n + 1/n goes to 1.
So L = 1/2.
4 Decide
L = 1/2 is less than 1: the series converges.
Careful!
L = 1 tells us nothing.
Only less than 1 converges, and more than 1 diverges.
5 Check
Let's check with a calculator: 10, 20, 50 terms add up to 5.8574, 5.9995, 6.
It settles.
Right!
Let's look back.
First: the next term.
Then: the ratio.
Then: the limit.
Then: decide.
Last: check.