Understanding Engineering Mathematics

Example

The generalnth degree polynomial (43
➤
) may be written in sigma notation as:

anxn+an− 1 xn−^1 +···+a 1 x+a 0 =

∑n

r= 0

arxr

Solution to review question 3.1.8

The series in full is:

∑^6

r= 1

r

r+ 1

=

1

1 + 1

+

2

2 + 1

+

3

3 + 1

+

4

4 + 1

+

5

5 + 1

+

6

6 + 1

=

1

2

+

2

3

+

3

4

+

4

5

+

5

6

+

6

7

Note that such series can be represented in many different ways in terms

of the sigma notation. For example the above series could be written just

as well as

∑^5

r= 0

r+ 1

r+ 2

3.2.9 Finite series

➤

89 109➤

The sigma notation can be used to write series, as for example in Section 3.2.8:

∑^7

r= 1

1

2 r−^1

=

1

20

+

1

21

+

1

22

+

1

23

+

1

24

+

1

25

+

1

26

= 1 +

1

2

+

1

4

+

1

8

+

1

16

+

1
32
The most well known and useful elementary series is the sum of ageometric progres-
sion(GP):

a,ar,ar^2 ,ar^3 ,...,arn−^1 ,...

whereais thefirst termandrthecommon ratio.Notice that thenth term isarn−^1.
Summing a finite geometric progression uses a nice argument. Let

Sn=a+ar+ar^2 +···+arn−^1

be the sum tonterms of suchaGP. Now multiply the series through byrto get

rSn=ar+ar^2 +ar^3 +···+arn

Then on subtracting the two series we obtain

Sn−rSn=( 1 −r)Sn=ar−arn