example of Taylor polynomials for the exponential function

Example 1.

We construct the nth Taylor polynomialMathworldPlanetmath for f(x)=ex around x=0. As we know all derivatives of ex equal ex and also, e0=1. Therefore, f(n)(0)=1 for any n. Thus:

T1(x) = 1+x
T2(x) = 1+x+x22
T3(x) = 1+x+x22+x33!=1+x+x22+x36
T4(x) = 1+x+x22+x33!+x44!=1+x+x22+x36+x424

In fact:


Comparison of ex with the Taylor pol. of deg. 1 (green), 2 (blue) and 3 (pink).

Let us use several Taylor polynomials to find approximations of the number e:

e = 2.718281828459045
eT1(1) = 1+1=2
eT2(1) = 1+1+1/2=2.5
eT3(1) = 1+1+1/2+1/6=8/3=2.6666¯
eT4(1) = 1+1+1/2+1/6+1/24=65/24=2.7083333¯
eT5(1) = 1+1+1/2+1/6+1/24+1/120=163/60=2.716666¯
Title example of Taylor polynomials for the exponential functionDlmfDlmfMathworldPlanetmath
Canonical name ExampleOfTaylorPolynomialsForTheExponentialFunction
Date of creation 2013-03-22 15:04:09
Last modified on 2013-03-22 15:04:09
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Example
Classification msc 41A58
Related topic LogarithmFunction
Related topic NaturalLogBase
Related topic EIsTranscendental
Related topic ExponentialFunction