Step by step
The question
Question: decide if the series converges: the sum of n/(2n² − 1) for n from 1 to infinity.
1 For big n
For big n, only the highest powers matter, top and bottom.
Cancel: about 1/(2n).
2 The known series
We know the sum of 1/n: a p-series with p = 1.
p is 1, so it diverges.
A number in front, 1/2, does not change that.
3 Compare
The bottom has a minus term: it is smaller than 2n².
A smaller bottom makes a bigger fraction: every term is bigger than 1/(2n).
The direct comparison test: our terms are bigger than terms whose sum grows for ever.
So ours grow for ever too.
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 3.15, 4.3 and 5.45.
It keeps growing.
Right!
Let's look back.
First: for big n.
Then: the known series.
Then: compare.
Then: decide.
Last: check.