8.6. Power Series http://www.ck12.org
Term-by-Term Integration of Power Series
Example 1 Find a power series forh(x) =tan−^1 xand its radius of convergence.
Solution. We recognizeh(x)as the antiderivative of 1 +^1 x 2 =∑∞n= 0 (− 1 )nx^2 n.
By Term-by-Term Theorem (B),h(x) =∑∞n= 0 ∫(− 1 )nx^2 ndx=∑∞n= 0 (−^12 )nn+x^21 n+^1 +Cand has radius of convergence 1.
ThenC=tan−^10 =0 andh(x) =∑∞n= 0 (−^12 )nn+x^21 n+^1.
Exercises
- Find a power series for ln( 1 +x^2 )and find the radius of convergence.
- Express∫tan−^1 xdxas a power series and find the radius of convergence.
- Find a power series for ln( 1 +x+x^2 )inxand inx+^12 , and find the radius of convergence.
Multimedia Links
For a video presentation showing how to differentiate and integrate power series(25.0), see Just Math Tutoring, Diff
erentiating and Integrating a Power Series (10:10).
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/619Series Multiplication of Power Series
Definition
(Series Multiplication) Thepower series productof two power series∑∞n= 0 anxnand∑∞n= 0 bnxnis a power
series∑∞n= 0 cnxndefined bycn=∑ni= 0 aibn−i=a 0 bn+a 1 bn− 1 +...+anb 0 (like polynomials).
A result is:the product of power series is the power series of the product.
If f(x) =∑∞n= 0 anxnandg(x) =∑∞n= 0 bnxnconverges on a common interval|x|<Rab, then their product power
series∑∞n= 0 cnxnalso converges onRaband is the power series for the product functionf(x)g(x).Example 1Find a power series for( 1 −x)(^11 − 2 x).
Solution. 1 −^1 x=∑∞n= 0 xnwith radius of convergence 1 and 1 −^12 x=∑∞n= 0 ( 2 x)nwith radius of convergence^12.
So for|x|<^12 ,