Taylor series for even functions and odd functions

Chapter 10, Problem 52E

(choose chapter or problem)

Problem 52E

Taylor series for even functions and odd functions (Continuation of Section 10.7, Exercise 55.) Suppose that  converges for all x in an open interval ( -R, R) Show that

a. If ƒ is even, then a1 = a3 = a5 = …… = 0, i.e., the Taylor series for ƒ at contains only even powers of x.

b. If ƒ is odd, then a0 = a2 = a4 = …… = 0 i.e., the Taylor series for ƒ at contains only odd powers of x.

Reference: Section 10.7, Exercise 55

Theory and Examples

Uniqueness of convergent power series

a. Show that if two power series )

b. Show that if  for all x in an open interval (-c, c) then an =0 for every n.