This disk is called the disk of convergence of the series, and R is called the radius of convergence.
We are now aiming to show that the power series E(x) is equal to the exponential function e^{x} for all real numbers x. (Remember that we have already defined the number e to be E(1).)
This result is really extremely deep (perhaps the deepest result so far in this class!) The fact that e^{x} can be expressed as a power series enables us to show that exponential functions (and logarithms) are differentiable and to calculate their derivatives.
Moreover, note that although e^{x} has, so far, only been defined for real x, the power series E(z) is defined for all complex numbers. Once we know that E(x)=e^{x} for all real x, it makes sense to define e^{z} as E(z) for all complex numbers. This then becomes the basis of our definition of the functions sin(x) and cos(x) , by means of the formulae:
The first big step to showing that E(x)=e^{x}, will be to show that the power series E(z) has the same multiplicative properties that we associate with exponential functions, namely that E(z)E(w)=E(z+w).