Taylor series 495

2

!

…

and multiplying

POLYNOMIAL

functions is relatively

straightforward. In 1715 English mathematician B

ROOK

T

AYLOR

worked to approximate complicated functions

with simple polynomials.

Suppose it is indeed possible to approximate a

complicated function f(x) as a polynomial:

f(x) ≈a0+ a1x+ a2x2+…+ anxn

One analyzes the situation by first noting that placing

x= 0 into this formula yields:

f(0) = a0+ 0 + 0 + … + 0

This shows that the approximation can be made exact,

at least at x= 0, by setting a0= f(0). To determine the

coefficient a1, differentiate once and then set x= 0:

f′(x) = 0 + a1+ 2a2x+ 3a3x2+ … + nanxn–1

f′(0) = a1

This shows that a1= f′(0) is a good choice. That is, by

setting a1to be this value, not only do the values of the

function and polynomial match at x= 0, but the slopes

of the two graphs also match at x= 0.

Differentiating another time and setting x= 0 (that

is, matching second derivatives) yields:

f′′(x) = 2a2+ 2 · 3a3x+ … + n(n– 1)anxn–2

f′′(0) = 2a2

and so . Continuing this way we obtain:

.

Thus a good approximation to the function f(x),at

least around the value x= 0, would be the polynomial:

The higher the degree the polynomial one uses, the bet-

ter the approximation would be. Thus the best polyno-

mial of all would be a polynomial of infinite degree,

that is, a

POWER SERIES

:

Mathematicians have proved that if fcan indeed be

differentiated infinitely many times, then this “approxi-

mation” is exact for the range of values the series con-

verges (called its

RADIUS OF CONVERGENCE

), that is, the

function really does equal the infinite sum expressed on

the right-hand side of the formula. This formula is

called a Taylor series.

As an example, consider the function f(x) = ex.

Differentiating and substituting in x= 0 yields:

f(x) = exf(0) = 1

f′(x) = exf′(0) = 1

f′′(x) = exf′′(0) = 1

and so . A study of power series

shows that this series has infinite radius of convergence,

and so this equation is valid for all values of x.

The Taylor series of f(x) = sin xis given by:

Similarly,

and

Since handheld calculators are programmed only to

add, subtract, multiply, and divide, Taylor series make

it possible to compute values of complicated functions.