http://www.ck12.org Chapter 8. Infinite Series
Term-by-Term Differentiation of Power Series
The goal of the next 3 sections is to find power series representations of certain classes of functions, namely
derivatives, integrals and products.
In the study of differentiation (resp. integration), we have found the derivatives (resp. integrals) of better known
functions, many with known power series representations. The power series representations of the derivatives (resp.
integrals) can be found by term-by-term differentiation (resp. integration) by the following theorem.
Theorem(Term-by-term differentiation and Integration)
Suppose∑∞n= 0 an(x−x 0 )nhas radius of convergenceRc. Then the functionfdefined byf(x) =∑∞n= 0 an(x−x 0 )nis
differentiable on(x 0 −Rc,x 0 +Rc)and
(A)f′(x) =∑∞n= 1 nan(x−x 0 )n−^1 ,
(B)∫f(x)dx=∑∞n= 0 an(x−n+x 10 )n+^1 +C,and these power series have same radius of convergenceRc.
(A) means (droppingx 0 )the derivative of a power series is the same as the term-by-term differentiation of the power
series:
dxd∑∞n=^0 anxn=∑∞n=^0 dxd(anxn)and
(B) meansthe integral of a power series is the same as the term-by-term integration of the power series:
∫ ∞
n∑= 0 (anxn)dx=∞
n∑= 0∫
(anxn)dx.Example 1Find a power series forg(x) =( 1 −^1 x) 2 and its radius of convergence.
Solution. We recognizeg(x)as the derivative of 1 −^1 xwhose power series representation is∑∞n= 0 xnwith radius of
convergenceRc=1. By (A),g(x) =dxd∑∞n= 0 xn=∑∞n= 1 nxn−^1 and has radius of convergence 1.
Exercises
Find a power series and the radius of convergence for the following functions:
- 2 −xx
- ( 1 −x^2 x) 3
- ( 1 −^2 xx) 3 +( 13 −xx^2 ) 4
Multimedia Links
For a video presentation of finding the interval of convergence of a power series(24.0), see Just Math Tutoring,
Power Series, finding the interval of convergence (9:47).
MEDIA
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