Math 152 – Spring 2016
Section 10.6
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Section 10.6 – Representations of Functions as
Power Series
1
can be rewritten as a power series since it is the sum of a geometric
The function 1−x
series with a = 1 and r = x.
∞
X
1
= 1 + x + x2 + x3 + x4 + · · · =
xn
1−x
n=0
Example 1. Rewrite the following functions as power series.
(a) f (x) =
1
1+x4
(b) f (x) =
1
3x+4
(c) f (x) =
x2
3x+4
for |x| < 1
Math 152 – Spring 2016
Section 10.6
Theorem. If the power series
∞
P
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cn (x − a)n has radius of convergence R > 0, then
n=0
the function defined by
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + · · · =
∞
X
cn (x − a)n
n=0
is differentiable (and therefore continuous) on the interval (a − R, a + R) and
1.
f 0 (x) = c1 + 2c2 (x − a) + 3c3 (x − a)2 + · · · =
∞
X
cn n(x − a)n−1
n=1
2.
Z
f (x) dx = C + c0 (x − a) + c1
∞
X
(x − a)2
(x − a)3
(x − a)n+1
+ c2
+· · · = C +
cn
2
3
n+1
n=0
The radii of convergence of the power series in Equations 1 and 2 above are both R.
Note.
1. The theorem says we can differentiate and integrate a power series by
differentiating and integrating each term, just like we would with a polynomial.
This is called term-by-term differentiation and integration. We can write
that as the following:
∞
∞
P
P
d
n
(a) dx
cn (x − a)
=
cn n(x − a)n−1
n=0
(b)
R
∞
P
n=0
cn (x − a)n
n=1
dx = C +
∞
P
n=0
n+1
cn (x−a)
n+1
2. This theorem only holds for power series.
3. When we integrate or differentiate a power series, the radius of convergence stays
the same, but the interval of convergence may not. After differentiation or integration, the endpoints of the integral may change from convergent to divergent
or vice versa.
Example 2. Use differentiation and integration to rewrite the following functions as
power series and find its radius of convegence.
(a)
1
(1−x)2
Math 152 – Spring 2016
(b)
Section 10.6
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3x2
(2+x2 )2
(c) ln(1 − x)
Note: In this example, we integrated to calculate the power series. We must
calculate C so our power series EXACTLY equals our function.
(d) x ln(7 + x4 )
Math 152 – Spring 2016
Section 10.6
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(e) g(x) = arctan(x)
(f) Differentiate the Bessel function we discussed in the previous section. Recall, the
radius of convergence for J0 (x) was R = ∞.
J0 (x) =
∞
X
(−1)n x2n
22n (n!)2
n=0
Math 152 – Spring 2016
Section 10.6
Example 3. (a) Evaluate
(b) Approximate
0.5
R
0
1
1+x7
R
1
1+x7
dx as a power series.
dx correct to within 10−7 .
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