Math 152 Class Notes
November 10, 2015
10.5 Power Series
A
power series
∞
X
is a series of the form
cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · ·
n=0
where x is a variable and the a and cn 's are constants. a is called the center and cn 's
are called the coecients of the series. In particular, if a = 0, the power series is of
the form
∞
X
cn xn = c0 + c1 x + c2 x2 + c3 x3 + · · ·
n=0
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) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · ·
whose domain is the set of all for x which the series converges. Thus a power series
can be viewed as a polynomial of innitely many terms.
Example 1. For what values of x is the series
∞
P
n=0
xn = 1+x+x2 +x3 +· · · convergent?
Example 2. For what values of x is the series
∞
P
n!xn convergent?
n=0
∞ xn
P
Example 3. For what values of x is the series
convergent?
n=0 n!
∞ (x − 2)n
P
convergent?
Example 4. For what values of x is the series
n
n=0
For a given power series
∞
P
cn (x − a)n , there are only three possibilities:
n=0
(a) The series converges only when x = a.
(b) The series converges for all x.
(c) There is a positive number R such that the series converges if |x − a| < R and
diverges if |x − a| > R.
The number R in case (c) is called the radius of convergence of the power series.
By convention, the radius of convergence is 0 in case (a) and ∞ in case (b).
The interval of convergence of a power series is the interval that consists of all
values of for which the series converges.
In general, the Ratio Test is be used to determine the radius of convergence. The
endpoints of the interval of convergence must be checked separately by other tests.
Example 5. Find the radius of convergence and interval of convergence of the series in
the Example 1-4.
(a)
∞
P
xn
n=0
(b)
∞
P
n!xn
n=0
∞ xn
P
(c)
n=0 n!
∞ (x − 2)n
P
(d)
n
n=0
Example 6. Find the radius of convergence and interval of convergence of the series
∞
X
(x − 3)n
√
n n+1
6
n=0
Example 7. Find the radius of convergence and interval of convergence of the series
∞
X
(3x − 2)n
n=0
Example 8. The series
∞
X
n=0
n2 + 4
cn xn converges at x = 2 and diverges at x = −4. Which of
the following series is certain to converge?
(a)
∞
X
cn (−5)n
n=0
(b)
∞
X
cn 4n
n=0
(c)
∞
X
cn 3n
n=0
(d)
∞
X
cn (−2)n
n=0
(e)
∞
X
n=0
cn