Taylor Series and its General Form

Khaled Ardah

May 21, 2013

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1 Introduction

The Taylor series is a representation of a function as an infinite sum of termscalculated from the values of its derivatives at a single point. If the seriesis centered at zero, the series is also called a Maclaurin series. Generally afinite number of terms are used instead of infinite number of terms to find theapproximate a function.

2 Motivation

In particular, the aim was to develop a method for representing a (differetiable)function f(x) as an (infinite) sum of powers of x. The main thought-processbehind the method is that powers of x are easy to evaluate, differentiate andintegrate, so by rewriting complicated functions as sums of powers of x we cangreatly simplify our analysis.

3 Single Variable Taylor Series

A one-dimensional Taylor series expansion of a real function f(x) about a pointx = a is given by:

f(x) = f(a) +f

′(a)

1!(x− a) +

f′′(a)

2!(x− a)2 + ... +

fn(a)

n!(x− a)n + ...

which can be written in the more compact sigma notation as:

f(x) =

∞∑n=0

fn(a)

n!(x− a)n

Note: If a = 0, the expansion is known as a Maclaurin series.

4 Taylor Series for Multiple Variables

To implement taylor series for a function which depents on more than one vari-able, four variables f(x,y,z,t) etc. First, we consider a point (a,b,c,d) wherewe implement the Taylor series. Now dx = a, dy = b, dz = c, dt = d.Then, we go for total derivative in finding the derivatives at the point. ie.,f′(a, b, c, d), f ′′(a, b, c, d), f ′′′(a, b, c, d), etc., as x, y, z, t are interdependent.

df =df

dxdx +

df

dydy +

df

dzdz +

df

dtdt

d2f =d(df)

dxdx +

d(df)

dydy +

d(df)

dzdz +

d(df)

dtdt

d3f =d(d2f)

dxdx +

d(d2f)

dydy +

d(d2f)

dzdz +

d(d2f)

dtdt

The derivative in general can be written as:

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dnf = [df

dxdx +

df

dydy +

df

dzdz +

df

dtdt]n

⇒ dnf = [d

dxdx +

d

dydy +

d

dzdz +

d

dtdt]nf(x, y, z, t)

Then;

f(x, y, z, t) = f(a, b, c, d) +1

1![d

dx(x− a) +

d

dy(y− b) +

d

dz(z− c) +

d

dt(t− d)]f+

1

2![d

dx(x− a) +

d

dy(y − b) +

d

dz(z − c) +

d

dt(t− d)]2f + ....+

1

n![d

dx(x− a) +

d

dy(y − b) +

d

dz(z − c) +

d

dt(t− d)]nf + ....

Then we can generalize the Taylor series to functions of more than onevariable with:

T (x1, ..., xk) =

∞∑n1=0

∞∑n2=0

....

∞∑nk=0

(x1 − a1)n1 ...(xk − ak)nk

n1!...nk!(dn1+...+nkf

dxn11 ...dxnk

k

)(a1, ..., ak)

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