e.
Term-by-Term Differentiationof PowerSeries
The goal of the next 3 sectionsis to find powerseriesrepresentationsof certainclassesof functions,namely
derivatives,integralsand products.
In the studyof differentiation(resp.integration),we havefoundthe derivatives(resp.integrals)of better
knownfunctions,manywith knownpowerseriesrepresentations.The powerseriesrepresentationsof the
derivatives(resp.integrals)can be foundby term-by-termdifferentiation(resp.integration)by the following
theorem.
Theorem(Term-by-termdifferentiationand Integration)
Suppose has radius of convergence R c. Then the function f defined by
is differentiableon (x 0 - Rc,x 0 +Rc) and(A)
(B) and thesepowerserieshavesameradiusof convergenceR
c.
(A) means(droppingx 0 the derivativeof a powerseriesis the sameas the term-by-termdifferentiationof
the powerseries:
and
(B) meansthe integralof a powerseriesis the sameas the term-by-termintegrationof the powerseries:
Example 1 Find a powerseriesfor and its radiusof convergence.
Solution.We recognizeg(x) as the derivativeof whosepowerseriesrepresentationis with
radiusof convergenceRc= 1. By (A), and has radiusof convergence