Understanding Engineering Mathematics

Here the inverse function is exactly the same as the original function.

There is nothing strange about this, but it does emphasise the fact that

the inverse of a function isnotthe same thing as the reciprocal of a

function!

3.2.8 Series and sigma notation

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89 109➤

We will say more about series (particularly infinite series) in Chapter 14. Here we want to
introduce the basic ideas by looking at two types of infinite series, thegeometricand the
binomial. In fact, these two are extremely important, in both the principles and practice
of series, and they occur frequently in applications in science and engineering.
Aseriesis a sum of asequence(i.e. a list) of terms which may be finite or infinite in
number, for example

2 + 4 + 8 + 16 + 32 ≡ 2 + 22 + 23 + 24 + 25

is afiniteseries. Dealing with finite series is essentially algebra.
Infinite series are usually implicitly defined by indicating that the series continues in a
particular way, for example:

1 +

1

2

+

1

4

+

1

8

+

1

16

+

1

32

+···+

1

2 n−^1

+···

↑

nth term

A useful shorthand for writing such series is thesigma notation. The Greek capital letter
sigma

∑
denotes summation. Thus, ifar (‘asubscriptr’) denotes some mathematical
object, such as a number or algebraic expression, then the expression

∑n

r=m

ar=am+am+ 1 +···+an− 1 +an

is ‘the sum of allarasrgoes frommton’.
The letterris called theindex of the summation, whilem,nare called thelower
andupper limitsof summation respectively. Note that the letterris in fact adummy
index– it can be replaced by any other:

∑n

r=m

ar=

∑n

i=m

ai

for example, in this respectris similar to the integration variable in definite integration
(Chapter 9):

∫b

a

f(x)dx=

∫b

a

f(t)dt