Using the f(x) = e3x with the center point (x0) at -9, we can start by computing the derivatives of f(x) since it is infinitely differentiable...
f(-9) = f0(-9) = e-27
f1(x) = f'(x) = 3 e3x
f1(-9) = 3e-27
f2(x) = f''(x) = 3 * 3 e3x
f2(-9) = 9e-27
We see the pattern and can now write the nth derivative as...
fn(x) = 3ne3x
f(x) = sum_[lb-->k=0]_[ub-->k=infinity] 3ke-27/k! * (x + 9)k