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 ex 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 ex 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 ex 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)=ex for all real x, it makes sense to define ez 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)=ex, 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).