Step by step
The question
Question: decide if the series converges: the sum of n/(n² + 1) for n from 1 to infinity.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The function
Make a function from the term: put x in place of n.
The series adds f(1), f(2), f(3) and so on.
For x from 1 on, it is positive.
And it gets smaller: the bottom grows faster than the top.
The integral test: the series and the area under f from 1 to infinity do the same thing.
Both are finite, or both are infinite.
Why: each term is a rectangle 1 wide and f(n) tall.
The rectangles hug the curve, so their total and the area grow together.
2 The integral
Infinity is not a number: stop at t, then let t grow, like improper integrals.
u-substitution: u = x² + 1.
Its derivative 2x dx is there: our top is 1/2 times 2x dx.
The integral is a log.
3 Let t go
Put in t, then take away the same with 1.
As t grows, this part grows without end.
So the area is infinite.
4 Decide
The area is infinite, so the series diverges too.
Careful!
The terms go to 0, but too slowly: the sum still grows for ever.
5 Check
Let's check with a calculator: 100 terms add up to about 4.52, 10000 terms to about 9.12.
Slowly, but it keeps growing.
Right!
6 Pick the choice
Our answer is diverges.
That is choice D.
7 Why not the others
A is converges.
Wrong: the terms go to 0, but that is not enough.
B is converges to 1/2.
Wrong: the terms go to 0, the sum keeps growing.
C is converges absolutely.
Wrong: it behaves like Σ 1/n.
Let's look back.
First: the function.
Then: the integral.
Then: let t go.
Then: decide.
Then: check.
Then: pick the choice.
Last: why not the others.