The Chemistry Maths Book, Second Edition

216 Chapter 7Sequences and series

EXAMPLE 7.14The exponential series

By Taylor’s theorem (fora 1 = 10 ),

where

for some point 01 < 1 b 1 < 1 x. Whenx 1 > 10 , the smallest and largest values ofR

n

are given

by

Then

For example, whenx 1 = 1 0.2andn 1 = 13 (and rounding all numbers to six decimal places),

the cubic approximation has value

and the error bounds are given by

0.000067 1 < 1 R

3

1 < 1 0.000067e

0.2

so that

1.221333 1 + 1 0.000067 1 < 1 e

0.2

1 < 1 1.221333 1 + 1 0.000067e

0.2

The lower bound is 1.221400. For the upper bound,

e

0.2

1 < 1 1.221333 1 + 1 0.000067e

0.2

e

0.2

1 − 1 0.000067e

0.2

1 = 1 0.999933e

0.2

1 < 1 1.221333

Therefore

e

0.2

1 < 1 1.221333 2 0.999933 1 = 1 1.221415

e

02

23

102

02

2

02

6

1 221333

.

≈+.+

.

• 
  

.

=.

() ()

1

21

1

2

212

++

!

++

!

• 
  

+!

<<++

!

++

+

x

xx

n

x

n

ex

xx

nn

x

n



() nn

x

n

e

n

x

!

• 
  

+!

+ 1

() 1

x

n

R

x

n

e

n

n

n

x

++

+!

<<

+!

11

() () 11

R

x

n

e

n

n

b

=

+!

+ 1

() 1

ex

xx

n

Rx

x

n

n

=+ +

!

++

!

1 +

2

2

 ()