Step by step
The question
Question: decide if the series converges: the sum of ((−1)ⁿn² + 1)/(2n²) for n from 1 to infinity.
1 The term
A series adds up the terms aₙ for ever.
First look at one term, ((−1)ⁿn² + 1)/(2n²).
The divergence test: if the terms do not go to 0, adding them for ever never settles.
The series diverges.
2 The limit
The size, without the sign.
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 size goes to 1/2.
With the sign, the terms jump between about 1/2 and minus that.
They do not go to 0.
3 Decide
The terms do not go to 0.
Adding numbers close to 1/2 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 partial sums keep jumping, −0.41, −0.91, −0.41, −0.91.
They never settle.
Right!
Let's look back.
First: the term.
Then: the limit.
Then: decide.
Last: check.