Step by step
The question
Question: find for which x the series converges: the sum of (x + 1)ⁿ over (n 4ⁿ).
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 + 1) to the n + 1, over (x + 1) to the n, leaves one (x + 1).
The powers of 4 leave 1/4.
The powers of n leave (n + 1) over n.
2 The limit
(n + 1)/n goes to 1.
So L is 1/4 times the size of x plus 1.
3 The radius
It converges when L is less than 1.
Multiply both sides by 4.
So the radius of convergence is 4: x can be up to 4 away from −1.
Careful!
Not 1/4!
L is 1/4 times the size, so the size must be less than 4.
Open up the bars: x is between −5 and 3.
4 The ends
At x = 3: (x + 1) is 4, and its powers cancel the powers of 4.
That is the sum of 1/n: it diverges.
At x = −5: 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
In interval notation: [−5, 3).
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, 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.