Clearly, from series we can compute function values. In this project we show that properties of functions can often be discovered from their Taylor or Maclaurin series. Using suitable series, prove the following.

(a) The formulas for the derivatives of e^z, cos z, sin z, cosh z, sinh z, and Ln (1 + z)

(b) 1/2(e^iz + e^-iz) = cos z

(c) sin z ≠ 0 for all pure imaginary z = iy ≠ 0