Math 1B
§ 11.8 Power Series
∞
∑c
A power series is a series of the form
n
x n = c 0 + c1 x + c 2 x 2 + c 3 x 3 + ...
n= 0
where x is a variable and the c n ’s are the coefficients of the series.
A power series may converge for some values of x and diverge for other values of x. The sum of the
€
series is a function
€ f x = c + c x + c x 2 + c x 3 + ...
( ) 0 1
2
3
whose domain is the set of all x for which the series converges.
∞
For example,€if c n = 1 we have the geometric series
∑x
n
= 1+ x + x 2 + x 3 + ...
n= 0
which converges if _______________ and diverges if _______________.
€
More generally, a series of the form
∞
∑ c (€x − a)
n
n
n= 0
is called a power series centered at a or a power series about a.
Example: For what values of x is each series convergent.
€
∞
a)
1
∑ n ( x − 5)
2
n=1
€
n
2
3
= c 0 + c1 ( x − a) + c 2 ( x − a) + c 3 ( x − a) + ...
∞
b)
€
n
(−1)
∑ 3n (n + 1) x n
n= 0
What happened in these two examples is typical:
∞
∑ c ( x − a)
A power series
n
n
about a converges on some symmetric interval about a. The interval may
n= 0
be:
Open:
€
Closed:
Half-open:
(a − R,a + R) or (−∞,∞)
[a − R,a + R] or [a,a]
(a − R,a + R] or [a − R,a + R)
€
€
This is called the interval of convergence of the series.
€
€
€
€
R = radius
of convergence
I = interval of convergence
(For (−∞,∞) , R = ∞ and for [ a,a] , R = 0 )
Series
€
€
Interval of Convergence
∞
Radius of Convergence
€
∑x €
n
n=0
∞
1
∑ n ( x − 5)
n
2
n=1
∞
n
(−1)
∑ 3n (n + 1) x n
n= 0
€
Example: Determine the radius of convergence and the interval of convergence for each series.
€
∞
a)
∑ n!(3x + 1)
n= 0
€
n
∞
b)
∑
( x − 6)
n
nn
n=1
€
∞
c)
n
∑ 4 ( x + 1)
n
n=1
€
n