Step by step
The question
Question: decide if the series converges: the sum of 1/n² for n from 1 to infinity.
1 The term
A series adds up the terms aₙ for ever.
First look at one term, 1/n².
The divergence test: if the terms do not go to 0, adding them for ever never settles.
The series diverges.
2 The limit
The bottom grows without end: the terms go to 0.
3 Decide
The terms go to 0.
Then this test cannot tell: some series like this converge, some diverge.
Careful!
Terms going to 0 is not enough!
The sum of 1/n has terms going to 0, but it diverges.
Two series, both with terms going to 0: one diverges and one converges.
Another test must decide: the integral test is next.
4 Check
Let's look with a calculator: the partial sums of the first 10, 100 and 1000 terms are 1.55, 1.63, 1.64.
Hard to tell from these alone: that is why we need another test.
Right!
Let's look back.
First: the term.
Then: the limit.
Then: decide.
Last: check.