Step by step
The question
Question: decide if the series converges: the sum of (−1)ⁿ⁺¹ over 2ⁿ 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.
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 1/2: 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 = 1/2.
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 0.3333, of 101 terms about 0.3333.
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.