# Some calculus with power series.

Theorem 10.4.1   Suppose that the power series is convergent on the interval . Its sum defines on that interval a function of the variable .

The function has derivatives of any order , and these derivatives are obtained by term-by-term differentiation, namely:     These power series have the same interval of convergence as thge original power series.

Theorem 10.4.2   Suppose that the power series is convergent on the interval and denote its sum by . Then the power series is convergent on the same interval and we have: Example 10.4.3 (A series for )   Let . This series is abolutely convergent for (v.s. Thm prop geometric series cv and Def def absolute cv). thus it is convergent; its sum is .

We integrate the function: Integrating the series term-by-term and equating both sides to 0 for , we have: Theorem 10.4.4 (The product of two power series)   Suppose that the two power series and are absolutely convergent for . Denote .

Then the series is absolutely convergent for and its sum is equal to ..

Example 10.4.5   Consider the two geometric series and , i.e. and The product of these series is given by: i.e. for even and for odd . We have: This could have been obtained either by a direct computation or by noting that Noah Dana-Picard 2007-12-28