Example: Determine the Taylor series for f(x) = e^x about x=0.

Solution

It is probably one of the easiest functions to get the Taylor series for. We just require recalling that,

f⁽ⁿ⁾(x) = e^x                                 n = 0,1,2,.....

Therefore we find,

f⁽ⁿ⁾(0) = 1                                  n = 0,1,2,.....

The Taylor series for this illustration is then,

Obviously, it's frequently easier to find the Taylor series about x=0 but we don't all the time do that.

Definition

A function, f(x), is termed as analytic at x=a if the Taylor series for f(x) regarding to x=a has a positive radius of convergence and converges to f(x).