The Chemistry Maths Book, Second Edition

196 Chapter 7Sequences and series

7.3 Finite series

Given a sequenceu

1

, u

2

, u

3

, =the partial sums

S

1

1 = 1 u

1

S

2

1 = 1 u

1

1 + 1 u

2

S

3

1 = 1 u

1

1 + 1 u

2

1 + 1 u

3

• 
  

with general term

(7.5)

also form a sequence,S

1

, S

2

, S

3

, =A sequence obtained in this way is called a series

and, when the sequence converges, the limit

(7.6)

is called the sum of the series.

The word series is commonly also used for the sums of terms themselves. A finite

seriesof nterms is then

with sumS

n

, and the infinite seriesis

with sum (if it exists) given by (7.6).

The arithmetic series

The sum of the first nterms of the arithmetic progression (7.1) is

The value of the finite series is obtained by considering the sum in reverse order,

S

n

1 = 1 [a 1 + 1 (n 1 − 1 1)d] 1 + 1 [a 1 + 1 (n 1 − 1 2)d] 1 +1-1+ 1 a

Sardaadad an

n

r

n

= +− =++ ++ +++−

=

∑

1

[( )] [ ][ ] [( 121  ))]d

r

r

uuuu

=

∑

=+++

1

123

∞



r

n

rn

uuuu u

=

∑

=++++

1

123



SS u

nn

n

r

n

r

==

















→→

=

∑

lim lim

∞∞

1

Suuu u u

nn

r

n

r

=++++ =

=

∑

123

1