Math 152 – Spring 2016
Section 10.5
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Section 10.5 – Power Series
Definition. A power series is a series of the form
∞
X
cn xn = c0 + c1 x + c2 x2 + c3 x3 + · · ·
n=0
The sum of the series is a function of x,
f (x) = c0 + c1 x + c2 x2 + c3 x3 + · · ·
The domain of the function f (x) is all x values for which the series converges. (For
each value of x, a power series is a series of constants that we can test for convergence.)
Example 1. Let cn = 1. Then the power series becomes the following geometric series:
∞
X
xn = 1 + x + x2 + x3 + · · · + xn + · · · =
n=0
1
1−x
which converges for −1 < x < 1 and diverges for x ≥ 1 or x ≤ −1.
Definition. A power series in (x − a) or a power series centered at a or a power
series about a is a series of the form
∞
X
cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · ·
n=0
Note.
• Power Series can be thought of as polynomials with infinitely many terms.
• For power series we often start the sum at n = 0.
• When x = 0 or (x − a) = 0 in the equations above, we use the convention that
x0 = 1 and (x − a)0 = 1.
• A power series centered at a will always converge if x = a (then all terms but the
first one are 0).
Example 2. Determine for which values of x the following series converge.
P (−1)n n! n
(a)
x
n
n=1
Math 152 – Spring 2016
(b)
P
n=1
Section 10.5
(x−4)n
n5n
Example 3. Find the domain of the Bessel Function of order 0 defined by
J0 (x) =
∞
X
(−1)n x2n
22n (n!)2
n=0
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Math 152 – Spring 2016
Section 10.5
Theorem. For a given power series
∞
P
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cn (x − a)n there are only three possibilities:
n=0
1. The series converges only when x = a.
2. The series converges for all x.
3. There is a positive number R such that the series converges if |x − a| < R and the
series diverges if |x − a| > R. (The series may converge or diverge when x − a = R
and x − a = −R.)
Definition. The radius of convergence for a power series
∞
P
cn (x − a)n is the num-
n=0
ber R such that the power series converges if |x − a| < R and diverges if |x − a| > R.
In cases 1. and 2. above, we say the radius of convergence is 0 and ∞, respectively.
The interval of convergence of a power series is the interval that consists of all values
of x for which the series converges. In Case 1 the interval of convergence is a single
point {a}, and in Case 2 the interval of convergence is (−∞, ∞). In Case 3, we can
rewrite |x − a| < R as a − R < x < a + R. The endpoints may or may not converge so
we have the following four possibilities for the interval of convergence
(a − R, a + R)
Series
∞
P
[a − R, a + R)
Radius of Convergence
(a − R, a + R]
[a − R, a + R]
Interval of Convergence
xn
n=1
P
n=1
P
n=1
(−1)n n! n
x
n
(x−4)n
n5n
(−1)n x2n
n=0 22n (n!)2
P∞
Note: The Ratio Test can be used to determine the radius of convergence for most
power series centered at a. However, the Ratio Test always fails when x is an endpoint
of the interval of convergence, so we must check the endpoints with some other test.
Math 152 – Spring 2016
Section 10.5
∞
P
Example 4. Suppose that
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cn xn converges when x = −3 and diverges when x = 5.
n=0
What can be said about the convergence or divergence of the following series?
(a)
∞
P
cn 2n
n=0
(b)
∞
P
n=0
cn
(c)
∞
P
(−1)n cn 7n
n=0
(d)
∞
P
cn 3n
n=0
Example 5. Find the radius and interval of convergence for the following power series.
(a)
∞
P
n=1
(−1)n xn
√
3 n
Math 152 – Spring 2016
(b)
∞
P
n=0
(c)
∞
P
n=0
(x+4)n
(2n−1)!
n(2x−3)n
3n
Section 10.5
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Math 152 – Spring 2016
(d)
∞
P
n=1
n3 (x−2)n
7·12·17···(5n+2)
Section 10.5
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