Step by step
The question
Question: decide if the series converges: the sum of (−1)ⁿ over (2n + 1) 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/(2n + 1).
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?
Divide top and bottom by n.
Every part with n at the bottom goes to 0.
So 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.
For big n, 1/(2n + 1) is like 1/n: that diverges, like the sum of 1 over n.
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 −0.2121, of 101 terms about −0.2171.
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.