The Chemistry Maths Book, Second Edition

202 Chapter 7Sequences and series

0 Exercises 36–39

Some finite series

The arithmetic series (7.8) is the simplest example of the family of series where

mis a positive integer; that is, the sum of powers of the natural numbers;

6

some of

these sums and other finite series are given in Table 7.1.

r

m

∑

=−

• 
  

=

• 
  

1

1

n 11

n

n

=++++













−++++

• 
  













1

1

1

2

1

3

11

2

1

3

11

1



nnn

r

n

r

n

r

n

rr r r

===

∑∑∑

• 
  

=−

• 
  

111

1

1

11

() 1

6

The general method for generating these sums is due to the Swiss mathematician Jakob Bernoulli

(1654–1705), Professor of mathematics at Basel. With his brother Johann (1667–1748), professor of mathematics

in Groningen and at Basel, and in collaboration with Leibniz, he also made contributions to the calculus, theory of

differential equations, series, and the calculus of variations. It was this collaboration that led to the success of

Leibniz’s formulation of the calculus; by 1700 most of the elementary calculus (described in this book) had been

developed. Newton’s method of fluxions was never well known outside England. It was Jakob Bernoulli who first

used the word ‘integral’.

Table 7.1 Some finite series

r

n

rr r n n

=

∑

++

=−

++

1

1

12

1

4

1

()( ) ()( )21 2

r

n

rr

n

n

=

∑

• 
  

=

• 
  

1

1

() 11

r

n

rr r nn n n

=

∑

++= ++ +

1

12

1

4

()( ) ()( )() 123

r

n

rr nn n

=

∑

+= + +

1

1

1

3

() ()( ) 12

r

n

rnnnnn

=

∑

=+ + + + = + + −

1

444 4543

12 3

1

5

1

2

1

3

1

30



=++= +

1

4

1

2

1

4

1

4

1

432 2 2

nnnnn()

r

n

rn

=

∑

=+ + + +

1

333 3

12 3

r

n

rnnnnnn

=

∑

=+ + + + = + + = +

1

222 232

12 3

1

3

1

2

1

6

1

6

 ()( 1 ()21n+

=+= +

1

2

1

2

1

2

1

2

nnnn()

r

n

rn

=

∑

=+ ++ +

1

123 

The sums of many other series can be derived from these.