I've tried looking through my book to see how to do these, but I just can't find it. Any help would be appreciated: 1) f(x) = 2x^5 - 3x^2 +2 2) f(x) = x^3 - x^7 3) f(x) = (1-x^2)/(1+x^2) 4) f(x) = 1/(x+2) Thanks in advance!
the definition of an even and an odd function is as follows: [tex] f(-x) = f(x) [/tex] is and even function and [tex] f(-x) = -f(x) [/tex] is an odd function.
It is also true (easy to prove) that a rational function (polynomial or quotient of polynomials) is even if and only if all exponents of x are even, odd if and only if all exponents of x are odd. Of course, functions don't always have "exponents"! sin(x) is an odd function and cos(x) is an even function.
But the series expansions precisely consist of only odd-numbered and only even-numbered polynomial terms, respectively. It's quite elegant.