E X Expansion Proof

After so much theory i finally prove that e x equals its maclaurin series.

E x expansion proof. The standard definition of an algebraic function is provided using an algebraic equation. In mathematics the taylor series is the most famous series that is utilized in several mathematical as well as practical problems. This is one of the properties that makes the exponential function really important. Find the taylor series expansion for e x when x is zero and determine its radius of convergence.

In the denominator for each term in the infinite sum. X 5 5. This leaves the terms x 0 n in the numerator and n. Complete solution before starting this problem note that the taylor series expansion of any function about the point c 0 is the same as finding its maclaurin series expansion.

We only needed it here to prove the result above. Allow for removal by moderators and thoughts about future. For a derivation of the formula for the maclaurin series of the exponential se. For real numbers c and d a function of the form is also an exponential function since it can be rewritten as.

X 3 3. X 4 4. As functions of a real variable exponential functions are uniquely characterized by the fact that the growth rate of such a function that is its derivative is directly. The derivative of e x is e x.

The taylor theorem expresses a function in the form of the sum of infinite terms. Where b is a positive real number not equal to 1 and the argument x occurs as an exponent. A taylor series is an expansion of some function into an infinite sum of terms where each term has a larger exponent like x x 2 x 3 etc. Featured on meta hot meta posts.

We can now apply that to calculate the derivative of other functions involving the exponential. These terms are determined from the derivative of a given function for a particular point. Now you can forget for a while the series expression for the exponential.