The Chemistry Maths Book, Second Edition

7.7 Approximate values and limits 215

If the series is terminated after a finite number of terms, the result is a polynomial

approximation to the function. The series therefore provides a sequence of such

approximations:

Some values of these are shown in Table 7.2.

Table 7.2 Values ofln(1+x)

u

1

1 = 1 xu

2

1 = 1 x 1 − 1 x

2

22 u

3

1 = 1 x 1 − 1 x

2

221 + 1 x

3

23 ln(1 1 + 1 x) 1 = 1 li

n→∞

m(u

n

)

00 0 0

0.0001 0.0000 9999 5 0.0000 9999 5000 0.0000 9999 5000

0.001 0.0009 995 0.0009 9950 0333 0.0009 9950 0333

0.01 0.0099 5 0.0099 5033 33 0.0099 5033 08

0.1 0.095 0.0953 333 0.0953 310

0.2 0.18 0.1826 66 0.1823 21

1.0 0.5 0.83 0.69

The table shows thatu

1

1 = 1 xis a good approximation to ln(1 1 + 1 x)whenx 1 ≤ 1 0.1and

that the series converges rapidly for these small values of x, each term providing at

least one additional figure of accuracy. Convergence is less good for larger values of x,

and eight terms are needed to give 10% accuracy whenx 1 = 11. The theoretical basis for

this use of the series is Taylor’s theorem.

Taylor’s theorem

Letf(x)be a continuous single-valued function of xwith continuous derivatives

f′(x), f′′(x), =, f

(n)

(x) in the interval ato x, and letf

(n+1)

(x) exist within the interval.

Then

1 f′′(a) 1 +1-

(7.25)

where

(7.26)

anda 1 < 1 b 1 < 1 xis some point in the interval. The termR

n

(x) is called the remainder

termand is the error involved in approximating the function by a polynomial of

degree n. The smallest and largest values ofR

n

(x) are lower and upper bounds to the

error. The infinite series is obtained in the limitn 1 → 1 ∞ifR

n

(x) 1 → 10 asn 1 → 1 ∞.

Rx

xa

n

fb

n

n

n

()

()

()

()

()

=

−

+!

+

+

1

1

1

• 
  

−

!

• 
  

()

() ()

()

xa

n

faRx

n

n

n

fx fa

xa

fa

xa

() ()

()

()

()

=+

−

!

′ +

−

12!

2

ux ux

x

ux

xx

12

2

3

23

223

=, = − , = − + ,