SERIES
A set of numbers, linked together by some rule is called a SEQUENCE.
2,4,6,8,10,…….
1,2,4,8,16,32,….
4,9,16,25,36,49,…..
the above are different types of sequence.
When the TERMS of a sequence are added, eg. 2+4+6+8+10, a SERIES
is obtained.
A FINITE SERIES stops after a finite number of terms.
3+6+9+12+15
is a finite series with 5 terms.
If the series does not stop but continues indefinitely it is AN INFITITE
SERIES.
1
1  12  14  18  161  321 .........  1024
 .....
Here the continuing dots indicate that there is no last term.
A finite series may also include dots but must end with a term eg.
1
1  12  14  18  161  321 .........  1024
is now a finite series.
A FINITE SERIES has a FINITE SUM. Ie it has a certain numerical
value.
AN INFINITE SERIES MAY CONVERGE TO A LIMIT
1
1  12  14  18  161  321 .........  1024
 .....
Here the consecutive terms in the series are becoming less significant
(decreasing in size) and so adding each consecutive term will lead to very
little change in the sum as we add more and more terms.
AN INFINITE SERIES MAY DIVERGE in which case no sum can be
obtained.
1  2  4  8 16  32  .....  2048  ....
Here the consecutive terms are increasing in value and so a significant
difference would be made by adding the next term. We must know where
to stop if this series is to have a sum.
THE SIGMA NOTATION
 is a symbol for “the sum of terms”
6
r
2

r 0
Substitute the
values
0,1,2,3,4,5,and 6
in place of the
parameter r and
then obtain their
sum
r takes values from
0 to 6 inclusive
6
2
So 
r 0
r
0
1
2
3
4
5
6
means 2  2  2  2  2  2  2
or 1  2  4  8  16  32  64
The sigma notation is particularly useful to express infinite series without
the need for dots. Eg.

(
r 0
1 r
2
is used to represent
)
1
1  12  14  18  161  321 .........  1024
 .....
Series that OSCILLATE between + and – terms can also be written in
this form by the use of (1)r . Eg
7
r 1
(

1)
 ( r 1 )
r 0
means
1  8  27  64 125
EXERCISE
WRITE THE FOLLOWING USING SIGMA NOTATION.
SERIES
1  8  27  64 125
2  4  6  8 10  ....  20
3  6  9  12  15  ....  99
1
2
 13  14  15  .........  501
1  13  19  271  ...........
8  4  2  1  12  14  .......
3  6  9 12 15 18  ......
MAKE SOME UP OF
YOUR OWN
SIGMA NOTATION
SERIES ARE NOT ALWAYS NUMERICAL
 THEY COULD BE ALGEBRAIC.
1  x  x 2  x3  x 4  ...... This is a series that could be written
in sigma notation as:

The series is
infinite so the
upper value is
infinity
r r
(

1)
x

x 0  1 gives the first
term. r=0 is the first
value of the parameter
r 0
Remember this gives the
oscillating effect of plus
,minus, plus minus, etc
A SPECIAL FAMILY OF SERIES ARE CALLED THE POWER
SERIES WHICH WILL BE DEALT WITH LATER IN THE COURSE.
x x 2 x3 x 4 x5
1       ........ is the POWER SERIES
1! 2! 3! 4! 5!
x
e
EXPANSION OF
.
IN SIGMA NOTATION THIS IS:

xr

r 0 r !
Remembering (if you knew already )that in mathematics the symbol !
means FACTORIAL. Ie repeated multiplication by decreasing
5!  5  4  3 2 1
And a fact is defined that 0!  1
consecutive integers. Eg