Step by step
The question
Question: decide if the series converges: the sum of (−1)ⁿ over 3ⁿ 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 = 1/3ⁿ.
The alternating series test: if the sizes get smaller and smaller and go to 0, the series converges.
2 Getting smaller?
Each term is the one before times 1/3: smaller.
3 Going to 0?
A fraction less than 1, to bigger and bigger powers, goes to 0.
4 Decide
Both checks pass: the series converges, by the alternating series test.
5 Absolutely or conditionally?
Now drop the signs: a geometric series with r = 1/3.
It converges.
The series converges even without its signs.
That is called converging absolutely.
So the series converges absolutely.
Careful!
Absolutely means it would converge even with all the signs plus.
6 Check
Let's check with a calculator: the sum of 100 terms is about −0.25, of 101 terms about −0.25.
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.