The Chemistry Maths Book, Second Edition

7.5 Tests of convergence 207

(ii) The exponential series

The general term is so that and. Then

and the series converges for all values of x.

0 Exercises 48, 49

Cauchy’s integral test

9

Leta

1

1 + 1 a

2

1 +1-1+ 1 a

r

1 +1-be a series of decreasing positive terms,a

r+ 1

1 < 1 a

r

. Leta(x)

be a function of the continuous variable xsuch thata(x) decreases as xincreases and

a(r) 1 = 1 a

r

. The series then

EXAMPLE 7.10The harmonic series.

In this case, and. Then

andln 1 x 1 → 1 ∞asx 1 → 1 ∞. The harmonic series therefore diverges.

0 Exercises 50, 51

Z

1

1

1

∞

∞

x

dx= x













ln

ax

x

()=

1

a

r

r

=

1

1

1

2

1

3

+++

diverges if Z diverges

1

∞

axdx()

converges if Z converges is

1

∞

axdx() ( f finite and unique)

lim lim

rr

a

a

x

r

r

r

→→

+

=

• 
  

=

∞∞

1

1

0

a

a

x

r

r

r

+

=

• 
  

1

1

a

x

r

r

r

+

+

=

+!

1

1

() 1

a

x

r

r

r

=

!

1

12 3

23

• 
  

!

• 
  

!

• 
  

!

• 
  

xx x



9

Augustin-Louis Cauchy (1789–1857). The leading French mathematician of the first half of the nineteenth

century, he is best known for his work on the theory of functions of a complex variable, with the Cauchy integral

theorem and the calculus of residues. He made contributions to, amongst others, partial differential equations, the

theory of elasticity, infinite series, and limits (see d’Alembert). He invented the word determinantfor his class of

alternating symmetric functions (1812). The integral test is sometimes named for MacLaurin.