Step by step
The question
Question: decide if the series converges: the sum of n/(2n + 1) for n from 1 to infinity.
1 The term
A series adds up the terms aₙ for ever.
First look at one term, n/(2n + 1).
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.
So the terms go to 1/2.
3 Decide
The terms do not go to 0.
Adding numbers close to 1/2 for ever never settles: the series diverges.
Careful!
1/2 is where each term ends up, not the sum.
The sum grows by about 1/2 every time.
4 Check
Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 4.41, 48.86, 498.28.
They keep growing.
Right!
Let's look back.
First: the term.
Then: the limit.
Then: decide.
Last: check.