The Chemistry Maths Book, Second Edition

7.8 Operations with power series 219

The numerator isx

2

(1 1 + 1 x 1 + 1 x

2

221 +1-). The denominator is

[(1 1 + 1 x 1 + 1 x

2

221 +1-) 1 − 1 1]

2

1 = 1 (x 1 + 1 x

2

221 +1-)

2

1 = 1 x

2

1 + 1 x

3

1 +1-

Therefore,

This example is important in the statistical mechanics of solids. In Einstein’s

theory of the heat capacity of simple atomic solids, each atom in the solid is

assumed to vibrate with the same frequency ν. The molar heat capacity of the

solid is then

withx 1 = 1 hν 2 kT, where his Planck’s constant, kis Boltzmann’s constant, Ris the

gas constant and Tis the temperature. The taking of the limitx 1 → 10 corresponds

to lettingT 1 → 1 ∞. ThenC

ν

1 → 13 R.

0 Exercises 73–77

7.8 Operations with power series

Let

be two power series with radii of convergenceR

A

andR

B

, respectively.

(i) Addition and subtraction.

The power series may be added or subtracted term by termto give a power series

(7.31)

whose radius of convergence is at least as large as the smaller ofR

A

andR

B

.

(ii) Multiplication.

The product ofA(x)andB(x) is the double infinite sum

Cx AxBx abx

rs

rs

rs

() ()()==

==

+

∑∑

00

∞∞

Cx Ax Bx a b x

r

rr

r

() () () (=±= ±)

=

∑

1

∞

Ax ax Bx bx

r

r

r

r

r

r

()==()

==

∑∑

00

∞∞

and

CR

xe

e

x

x

ν

=

−

3

1

2

2

()

xe

e

xx

xx

x

x

x

2

2

2

2

1

1

1

10

()

()

− ()

=

++

++

→→





as