Step by step
The question
Question: decide if the series converges: the sum of 2n/(n² + 9) for n from 1 to infinity.
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.
2 The integral
Infinity is not a number: stop at t, then let t grow, like improper integrals.
u-substitution: u = x² + 9.
Its derivative 2x dx is there, up to the number 1.
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 7.01, 10000 terms to about 16.2.
Slowly, but it keeps growing.
Right!
Let's look back.
First: the function.
Then: the integral.
Then: let t go.
Then: decide.
Last: check.