Calculus- Early Transcendentals, 2021a

364 Sequences and Series
|x + 2| < 3
−3 < x + 2 < 3
−5 < x < 1.
So the interval of convergence for this series is (−5,1). The center of this interval is at −2, which is at
distance 3 from the endpoints, so the radius of convergence is 3, and we say that the series is centered at
−2.
Interestingly, if we compute the sum of the series we get
∞
∑
n=0
Multiplying both sides by 1/3 we obtain
( ) x + 2 n
=
3
1
1 − x+2
3
= 3
1 − x .
∞
(x + 2)
∑
n
n=0
3 n+1 = 1
1 − x ,
which we recognize as being equal to
∞
∑ x n ,
n=0
so we have two series with the same sum but different intervals of convergence.
This leads to the following definition:
Definition 9.48: Power Series
A power series centered at c has the form
where c and each a n are real numbers.
∞
∑ a n (x − c) n ,
n=0
Exercises for 9.8
Exercise 9.8.1 Find the radius and interval of convergence for each series. In part c), do not attempt to
determine whether the endpoints are in the interval of convergence.
(a)
(b)
(c)
∞
∑ nx n
n=0
∞
x
∑
n
n=0
n!
∞
∑
n=1
(d)
n!
(x − 2)n
nn (e)
∞
∑
n=1
(n!) 2
(x − 2)n
nn ∞
(x + 5)
∑
n
n=1
n(n + 1)