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10.5 Power Series
Do you like polynomials? Well, then you’ll love power series. A power series is an infinite degree polynomial.
A power series is a series of the form:
∞
X
cn xn = c0 + c1 x + c2 x2 + c3 x3 + · · ·
n=0
More generally, A power series centered at a is a polynomial of the form:
∞
X
cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · ·
n=0
The first power series given above is just a power series centered at 0.
Depending on the values of x, a power series may converge to a finite sum for some values of x and diverge
for other values of x. The interval of x-values for which the series converges is called the interval of
convergence.
Notice that for ALL power series, when x = a the series will definitely converge since you have a sum of
zeros, but for other values of x, the convergence will depend on what the cn ’s are. So every power series at
least converges at its center.
Theorem: For a given power series
convergence.
∞
P
n=0
cn (x − a)n , there are ONLY three possibilities for the interval of
(I) The series converges ONLY at x = a.
(II) The series converges for ALL x.
(III) There is a positive number R such that the series converges when |x − a| < R and diverges when
|x − a| > R. In other words, there is a symmetric interval centered at a so that the series converges for all
values of x inside this interval. However, this does NOT tell us what happens when |x − a| = R. The series
may or may not converge at the endpoints of the interval. You have to check the endpoints separately.
The number R is called the radius of convergence. What is the radius of convergence for cases I and II?
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Math 152, Benjamin
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Examples: Determine the radius and interval of convergence for the following power series.
∞
X
5n xn
n=0
n!
∞
X
xn
n(−6)n
n=1
2
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∞
X
3n(x − 2)n
n=1
5n−1
∞
X
(2n)!(x + 6)n
n=0
n+4
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∞
X
(−4)n (2x − 5)n
n=0
n2 + 5
4
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∞
X
(3x + 1)n
n=2
7n ln n
Suppose we know that the series
series converge?
∞
X
cn 8n
∞
X
cn (−6)n
∞
X
cn
∞
X
(−1)n cn 4n
∞
X
cn xn converges for x = 4 and diverges for x = −7. Do the following
n=0
n=0
n=0
n=0
n=0
5
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Suppose it is known that the series
∞
X
an (x − 2)n converges when x = 5. On what interval can we guarantee
n=0
convergence based on this information?
10.6 Representations of Functions as Power Series
Recall that the sum of a geometric series,
∞
P
n=0
So, the power series
∞
P
n=0
xn =
ar n is
a
as long as |r| < 1.
1−r
1
as long as |x| < 1. The radius of convergence is 1 and the interval of
1−x
convergence is (−1, 1).
Find a power series representation for the following functions. What is the radius of convergence for each?
4
f (x) =
1 − 5x
f (x) =
x
7−x
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f (x) =
2x5
4 + x2
Find the sum of the series
∞
X
2n x3n
n=0
4(−5)n
. What is the radius of convergence?
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We can differentiate and integrate power series by remembering that a power series is just an infinite degree
polynomial.
Given f (x) =
∞
P
n=0
(1) f ′ (x) =
∞
P
n=1
(2)
Z
cn xn = c0 + c1 x + c2 x2 + c3 x3 + · · ·,
ncn xn−1 = c1 + 2c2 x + 3c3 x2 + · · ·
f (x) dx = C +
∞
X
cn xn+1
n=0
n+1
= C + c0 x +
c1 x2 c2 x3
+
+ ···
2
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Imnportant Notes;
• When differentiating a power series, the starting index can increase by one if the first term of the original
series was a constant.
• If the radius of convergence of the original power series is R, then it remains R after differentiating or
integrating.
• However, the interval of convergence may change, meaning that convergence at the endpoints may change.
Find power series centered at 0 for the following functions. Determine the radius of convergence for each.
f (x) =
1
(5 + x)2
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f (x) =
x
(3 − 5x)2
f (x) = ln(1 + x)
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f (x) = x2 ln(3 − x7 )
f (x) = arctan x
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Integrate
Z
Evaluate
Z
0
x3 arctan(7x2 ) dx as a power series.
1/2
x4
1
dx as a series.
+1
Approximate this integral correctly to within an error of 0.0001.
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