Step by step
The question
Question: find for which x the series converges: the sum of xⁿ over 2ⁿ.
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 2 leave 1/2.
2 The limit
So L is 1/2 times the size of x.
3 The radius
It converges when L is less than 1.
Multiply both sides by 2.
So the radius of convergence is 2: x can be up to 2 away from 0.
Careful!
Not 1/2!
L is 1/2 times the size, so the size must be less than 2.
Open up the bars: x is between −2 and 2.
4 The ends
At x = 2: x is 2, and its powers cancel the powers of 2.
The terms do not go to 0: it diverges.
At x = −2: the same, but with signs that take turns.
The terms do not go to 0: it diverges.
5 The interval
In interval notation: (−2, 2).
A square bracket means the end is in; a round one, it is not.
6 Check
Let's check an x inside, 1, with a calculator: 40 terms and 80 terms add up to almost the same, 1.
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.