Verify for yourself that:

So the Maclaurin series of is given by:

This series converges for all (use the ratio test to verify).

To prove that is equal to the sum of its Maclaurin series, we must show that where is the remainder, i.e.:

where

This fact should come in handy:

With this, we've shown is equal to its Taylor series over the reals and thus analytic.