Step by step
The question
Question: decide if the series converges: the sum of 6n² over (3n² + n) for n from 1 to infinity.
1 The term
A series adds up the terms aₙ for ever.
First look at one term, 6n² over (3n² + 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 to the 2, the highest power at the bottom.
Every part with n at the bottom goes to 0.
So the terms go to 2.
3 Decide
The terms do not go to 0.
Adding numbers close to 2 for ever never settles: the series diverges.
Careful!
2 is where each term ends up, not the sum.
The sum grows by about 2 every time.
4 Check
Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 18.32, 196.84, 1995.31.
They keep growing.
Right!
Let's look back.
First: the term.
Then: the limit.
Then: decide.
Last: check.