The Theory Behind Taylor Series

The Theory Behind Taylor Series Swapnil Sunil Jain April 23, 2006

The Theory Behind Taylor Series We first make a safe assumptionPlanetmathPlanetmath that any function f(x) can be approximated with the help of an nth degree polynomialPlanetmathPlanetmath p(x). Thus,

f(x)p(x) =anxn+an-1xn-1+...+a4x4+a3x3+a2x2+a1x+a0 (1)

Now all we need to do in order to know this function is to figure out the values of the coefficients an,an-1,...,a4,a3,a2,a1,a0 of the polynomial p(x).

Getting a0 is easy, we just set x = 0, and we get a0=p(0). However, getting the rest of the coefficients requires some trick. But before doing anything else, let’s just take the first few derivatives of the polynomial p(x):

p(x) = nanxn-1+...+4a4x3+3a3x2+2a2x+a1
p′′(x) = n(n-1)anxn-2+...+43a4x2+32a3x+2a2
p3(x) = n(n-1)(n-2)anxn-3+...+432a4x+32a3
p4(x) = n(n-1)(n-2)(n-3)anxn-4+...+432a4

Now getting the next coefficient a1 is easy! We just set x = 0 and we get p(0)=a1. Similarly, setting x = 0 for all the rest of the derivatives we get:

p′′(0) = 2a2
p3(0) = 32a3
p4(0) = 432a4

Hmmm… Do you see a pattern? The coefficient in front of the nth term seems to be n!. By this logic, the nth derivative of p(x) (evaluated at 0) should look like the following:


Solving for an, we get:


By using this formulaMathworldPlanetmathPlanetmath we can figure out all the coefficients (i.e. an,an-1,...,a2,a1,a0) of the polynomial p(x) (and be able to approximate f(x)!). Thus, our original function (1), which was approximated by the polynomial p(x), could be written (in the form of a polynomial) as:


or in summation form as:

f(x)k=0nfk(0)k!xk (2)

This is the nth-order Taylor seriesMathworldPlanetmath expansion of f(x) about the point x = 0. However, this is only an approximation. To get the exact form, we have to have an infinite series summing over all the possible integer values of k from 0 to infinityMathworldPlanetmathPlanetmath (i.e. an infiniteMathworldPlanetmath degree polynomial)

f(x)=k=0fk(0)k!xk (3)

Now, one more thing that we can do is to change the point of evaluation from x = 0 to x = a for some real value a. Thus, the Taylor series expansion of f(x) about a point x = a becomes the following (Note that setting a = 0 gives us back our above expression):

f(x)=k=0fk(a)k!(x-a)k (4)
Title The Theory Behind Taylor Series
Canonical name TheTheoryBehindTaylorSeries1
Date of creation 2013-03-11 19:24:28
Last modified on 2013-03-11 19:24:28
Owner swapnizzle (13346)
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