Step by step
The question
Question: decide if the series converges: the sum of (n + 2)/(n² + 5n) for n from 1 to infinity.
1 For big n
For big n, only the highest powers matter, top and bottom.
Cancel: about 1 over n.
2 The known series
We know the sum of 1/n: a p-series with p = 1.
p is 1, so it diverges.
3 Compare
The terms are not simply smaller or bigger here.
So use the limit comparison test: divide aₙ by bₙ and let n grow.
Dividing by 1/n is the same as times its flip.
Top and bottom now have the same highest power with the same number: the limit is 1.
The limit is a positive number.
Then the two series do the same thing: both converge, or both diverge.
4 Decide
So our series diverges.
Careful!
Smaller than a series that diverges tells us nothing.
We need bigger than a diverging one, or the limit test.
5 Check
Let's check with a calculator: 100, 1000 and 10000 terms add up to about 3.85, 6.12 and 8.42.
It keeps growing.
Right!
Let's look back.
First: for big n.
Then: the known series.
Then: compare.
Then: decide.
Last: check.