CK12 Calculus - Single Variable

(Marvins-Underground-K-12) #1

Exercise
Find a powerseriesand the radiusof convergencefor the followingfunctions:











  1. 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



  1. Find a powerseriesforln(1 +x^2 ) and find the radiusof convergence.

  2. Express as a powerseriesand find the radiusof convergence.

  3. 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.

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