We first derive the Taylor series for
*f* (*x*) = *e*^{x}*a* = 0.

Since for every natural number *k*, the
*k*-th*f*^{ (k)}(*x*) = *e*^{x},

Therefore the Taylor series is

Now take any real number *x*. We will use the
Lagrange
estimate of the remainder to prove that this Taylor series converges to
the exponential at this *x*. If we denote by *I* the closed
interval with endpoints 0 and *x*, we have the estimate

Now we need to ask where *x* is. If
*x* = 0,*T*(0) = 1 = e^{0}.*x* < 0,*I* is
*x*,0],*e*^{0} = 1.

We used the fact that factorial grows faster than geometric sequence at
infinity, see the
scale of powers.
So for negative *x* we have the convergence of
*T*(*x*)*e*^{x}

When *x* > 0,*I* is
*x*]*x*.
Therefore

So also for positive *x* we have the convergence.

Note that if we fix some interval
*A*,*A*],*x* from this interval.

By the same argument as above, this error estimate goes to 0 as *N*
goes to infinity, which proves that the Taylor series for exponential
converges uniformly to *e*^{x}*A*,*A*].