Step by step
The question
Question: find for which x the series converges: the sum of xⁿ over n².
1 The ratio
Use the ratio test.
x can be negative, so we take sizes, with bars.
Divide the next term by this one.
x to the n + 1, over x to the n, leaves one x.
The powers of n leave (n + 1) over n.
2 The limit
(n + 1)/n goes to 1.
So L is the size of x.
3 The radius
It converges when L is less than 1.
So the radius of convergence is 1: x can be up to 1 away from 0.
Open up the bars: x is between −1 and 1.
4 The ends
At x = 1: x is 1, so x to the n is just 1.
A p-series with p = 2: it converges.
At x = −1: the same, but with signs that take turns.
A p-series with p = 2: it converges, even with the signs.
Careful!
The ratio test says nothing at the ends, where L is exactly 1.
Each end must be tested on its own.
5 The interval
In interval notation: [−1, 1].
A square bracket means the end is in; a round one, it is not.
6 Check
Let's check an x inside, 0.5, with a calculator: 40 terms and 80 terms add up to almost the same, 0.5822.
It converges there.
Right!
Let's look back.
First: the ratio.
Then: the limit.
Then: the radius.
Then: the ends.
Then: the interval.
Last: check.