The exponential function e^x has a Taylor series expansion given by the formula,
The above formula is the Taylor series for e^x around the point x=0. It is also known as the Maclaurin series and it expresses the exponential function in the form of a “polynomial of infinite degree”.
The infinite series on the right-hand side means that the value of the infinite series at a point converges to the value of the exponential function at that point.
This series is valid for all real numbers, that is, the series on the right-hand side is convergent for all real values of x. In fact, this series is sometimes taken to be the definition of the exponential function.
Consider the function f(x) = e2x. We can expand it in the form of a Taylor series using the above formula by substituting 2x instead of x. We have that,
Derivation of e^x Taylor Series:
We now give a heuristic proof/derivation for the Taylor series formula. We begin by assuming that the exponential function can be expanded in the form of an infinite polynomial as shown below.
ex = a0 + a1x + a2x2 +…..+ anxn + ….
We need to determine the coefficients ai of the above “infinite degree polynomial”.
Putting x=0 on both sides in the above series we get that a0=1.
Let us assume that the above series can be differentiated term by term. Differentiating once on both sides we get,
Once again substituting x=0 on both sides, the higher order terms vanish and we get that a1=1.
Similarly, on differentiating the series ‘n’ number of times, we get,
Once again substituting x=0 on both sides, the higher order terms vanish and we get,
an = 1/n!
This “proves” the formula for the Taylor series of e^x. The above proof is merely a heuristic proof and not rigorous because we have simply assumed that the function can be expanded as a polynomial of an infinite degree.
We also assumed that the term-by-term differentiation of an infinite series is a valid operation.
e^x Taylor Series about the point x=a:
We have seen above the formula for the Taylor series of e^x about the point x=0. The formula for the Taylor series of e^x about the point x=a is given as,