Step by step
The question
Question: find for which x the series converges: the sum of (x − 2)ⁿ over 3ⁿ.
A series with x in it: for which x does it converge?
There are four choices.
First we solve it, the normal way.
Then we pick.
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 − 2) to the n + 1, over (x − 2) to the n, leaves one (x − 2).
The powers of 3 leave 1/3.
2 The limit
So L is 1/3 times the size of x minus 2.
3 The radius
It converges when L is less than 1.
Multiply both sides by 3.
So the radius of convergence is 3: x can be up to 3 away from 2.
Careful!
Not 1/3!
L is 1/3 times the size, so the size must be less than 3.
Open up the bars: x is between −1 and 5.
4 The ends
At x = 5: (x − 2) is 3, and its powers cancel the powers of 3.
The terms do not go to 0: it diverges.
At x = −1: the same, but with signs that take turns.
The terms do not go to 0: it diverges.
5 The interval
−1 is not in: an open dot.
5 is not in: an open dot.
Shade the x's in between.
In interval notation: (−1, 5).
A square bracket means the end is in; a round one, it is not.
6 Check
Let's check an x inside, 3.5, with a calculator: 40 terms and 80 terms add up to almost the same, 1.
It converges there.
Right!
7 Pick the choice
Our answer is (−1, 5).
That is choice C.
8 Why not the others
A is [-1, 5].
Wrong: at the ends the terms do not go to 0.
B is (−3, 3).
Wrong: the center is 2, not 0.
D is (1, 3).
Wrong: the radius is 3, not 1.
Let's look back.
First: the ratio.
Then: the limit.
Then: the radius.
Then: the ends.
Then: the interval.
Then: check.
Then: pick the choice.
Last: why not the others.