3 Series

A series is the sum of the terms of a sequence. For example, the harmonic series is

1 + 1 2 + 1 3 + 1 4 +

and the alternating harmonic series is

1 1 2 + 1 3 1 4 +

3.1 The summation notation

If we consider a general sequence

a 1 , a 2 , , a n ,

then the sum of the first k terms a 1 + a 2 + a 3 + + a k is concisely denoted by p = 1 k a p .

That is,

a 1 + a 2 + a 3 + + a k = p = 1 k a p

When we encounter the expression p = 1 k a p we let the index ‘ p ’ in the term a p take, in turn, the values 1 , 2 , , k and then add all these terms together. So, for example

p = 1 3 a p = a 1 + a 2 + a 3 p = 2 7 a p = a 2 + a 3 + a 4 + a 5 + a 6 + a 7

Note that p is a dummy index; any letter could be used as the index. For example i = 1 6 a i , and m = 1 6 a m each represent the same collection of terms: a 1 + a 2 + a 3 + a 4 + a 5 + a 6 .

In order to be able to use this ‘summation notation’ we need to obtain a suitable expression for the ‘typical term’ in the series. For example, the finite series

1 2 + 2 2 + + k 2

may be written as p = 1 k p 2 since the typical term is clearly p 2 in which p = 1 , 2 , 3 , , k in turn.

In the same way

1 1 2 + 1 3 1 4 + 1 16 = p = 1 16 ( 1 ) p + 1 p

since an expression for the typical term in this alternating harmonic series is a p = ( 1 ) p + 1 p .

Task!

Write in summation form the series

1 1 × 2 + 1 2 × 3 + 1 3 × 4 + + 1 21 × 22

First find an expression for the typical term, “the p t h term”:

a p = 1 p ( p + 1 )

Now write the series in summation form:

1 1 × 2 + 1 2 × 3 + + 1 21 × 22 = p = 1 21 1 p ( p + 1 )

Task!

Write out all the terms of the series p = 1 5 ( 1 ) p ( p + 1 ) 2 .

Give p the values 1 , 2 , 3 , 4 , 5 in the typical term ( 1 ) p ( p + 1 ) 2 :

1 2 2 + 1 3 2 1 4 2 + 1 5 2 1 6 2 .