Exercise
Find a powerseriesand the radiusof convergencefor the followingfunctions:
- Term-by-Term Integrationof PowerSeries
Example1 Find a powerseriesforh(x) =tan-1xand its radiusof convergence.
Solution.We recognizeh(x) as the antiderivativeof.
By Term-by-Term Theorem(B), and has radius
of convergence1.
ThenC=tan-10 = 0 and.
Exercise
- Find a powerseriesforln(1 +x^2 ) and find the radiusof convergence.
- Express as a powerseriesand find the radiusof convergence.
- Find a powerseriesforln(1 +x+x^2 ) inxand in , and find the radiusof convergence.
SeriesMultiplicationof PowerSeries
Definition(SeriesMultiplication)Thepowerseriesproductof two powerseries and
isapowerseries definedby (likepolynomials).
A resultis:the productof powerseriesis the powerseriesof the product.