Step by step
The question
Question: find for which x the series converges: the sum of (x − 1)ⁿ over (n 2ⁿ).
A series with x in it: for which x does it converge?
1 The ratio
The terms have powers of x: the ratio test handles powers well.
L less than 1 converges, more than 1 diverges, exactly 1 cannot tell.
x can be negative, so we take sizes, with bars.
Divide the next term by this one.
(x − 1) to the n + 1, over (x − 1) to the n, leaves one (x − 1).
The powers of 2 leave 1/2.
The powers of n leave n/(n + 1).
2 The limit
n/(n + 1) goes to 1.
So L is 1/2 times the size of x minus 1.
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 1.
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 −1 and 3.
4 The ends
At x = 3: (x − 1) is 2, and its powers cancel the powers of 2.
That is the sum of 1/n: it diverges.
At x = −1: the same, but with signs that take turns.
The signs take turns and the sizes 1/n go to 0: it converges, by the alternating series test.
5 The interval
−1 is in: a filled dot.
3 is not in: an open dot.
Shade the x's in between.
In interval notation: [−1, 3).
A square bracket means the end is in; a round one, it is not.
6 Check
Let's check an x inside, 2, with a calculator: 40 terms and 80 terms add up to almost the same, 0.6931.
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.