Step by step
The question
Question: decide if the series converges: the sum of (n² + 1)/n for n from 1 to infinity.
1 The term
A series adds up the terms aₙ for ever.
First look at one term, (n² + 1) over n.
The divergence test: if the terms do not go to 0, adding them for ever never settles.
The series diverges.
2 The limit
Divide top and bottom by n, the highest power at the bottom.
Every part with n at the bottom goes to 0.
The top still grows: the terms grow without end.
3 Decide
The terms do not go to 0.
Adding numbers that grow for ever never settles: the series diverges.
Careful!
The limit of the terms tells us only about the terms, not about the sum.
4 Check
Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 57.93, 5055.19, 500507.49.
They keep growing.
Right!
Let's look back.
First: the term.
Then: the limit.
Then: decide.
Last: check.