Step by step
The question
Question: decide if the series converges: the sum of (−1)ⁿ⁺¹ over n for n from 1 to infinity.
1 The size
Write the first terms: the signs take turns.
Take the size of each term, without the sign: b n = 1 over n.
The alternating series test: if the sizes get smaller and smaller and go to 0, the series converges.
2 Getting smaller?
n + 1 is bigger than n, so the bottom gets bigger: each size is smaller than the one before.
3 Going to 0?
The bottom grows without end: the sizes go to 0.
4 Decide
Both checks pass: the series converges, by the alternating series test.
5 Absolutely or conditionally?
Now drop the signs: the sum of 1/n is a p-series with p = 1.
p is not bigger than 1: it diverges.
The series converges only because the signs take turns.
That is called converging conditionally.
So the series converges conditionally.
Careful!
The signs make a difference!
Without them it diverges, but with them taking turns, it converges.
6 Check
Let's check with a calculator: the sum of 100 terms is about 0.6882, of 101 terms about 0.6981.
They jump less and less, and the sum is between them.
It settles.
Right!
Let's look back.
First: the size.
Then: getting smaller?.
Then: going to 0?.
Then: decide.
Then: absolutely or conditionally?.
Last: check.