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.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
Why: the sum steps forward, then back a bit less, then forward a bit less.
It zig-zags in a window that shrinks to one point.
2 Do the sizes get smaller
The next size puts n + 1 in place of n.
A bigger n makes a bigger bottom: each size is smaller than the one before.
3 Do they go 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 over √n is a p-series with p = 1/2.
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 0.555, of 101 terms about 0.6545.
They jump less and less, and the sum is between them.
It settles.
Right!
7 Pick the choice
Our answer is converges conditionally.
That is choice B.
8 Why not the others
A is converges absolutely.
Wrong: Σ 1/√n diverges: p = 1/2 is not more than 1.
C is diverges.
Wrong: the alternating series test works: 1/√n shrinks to 0.
D is converges to 0.
Wrong: the sum is not 0.
Let's look back.
First: the size.
Then: do the sizes get smaller.
Then: do they go to 0.
Then: decide.
Then: absolutely or conditionally.
Then: check.
Then: pick the choice.
Last: why not the others.