Step by step
The question
Question: decide if the series converges: the sum of 2ⁿ over n factorial for n from 1 to infinity.
For terms with powers like 2 to the n, and factorials.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 + 1, over 2 to the n, is just 2.
(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 (n + 1)!
over n!
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 6.389, 6.3891, 6.3891.
It settles.
Right!
6 Pick the choice
Our answer is converges.
That is choice C.
7 Why not the others
A is diverges.
Wrong: n!
grows faster than 2ⁿ: the ratio goes to 0.
B is diverges to infinity.
Wrong: the terms shrink fast.
D is cannot be decided by the ratio test.
Wrong: the ratio's limit is 0, less than 1.
Let's look back.
First: the next term.
Then: the ratio.
Then: the limit.
Then: decide.
Then: check.
Then: pick the choice.
Last: why not the others.