Understanding Engineering Mathematics

Exercise on 14.6

Adapt the argument of this section to show that the series

1 +

1

√

2

+

1

√

3

+

1

√

4

+···+

1

√

r

+···

does not add up to a finite quantity, i.e. it diverges.

14.7 Infinite sequences

As a prelude to infinite series we look at infinitesequences(102

➤
), which are simply lists
that continue indefinitely – adding terms of an infinite sequence produces an infinite series.
Such sequences arise naturally in such infinite processes asiteration, where we continually
improve an approximation by recycling it until we get the accuracy
we want.
An ordered set of numbers:

u 1 ,u 2 ,u 3 ,...,un,...

is called asequenceorinfinite sequence.unis called thenth term(105

➤
). Thenth
term may be defined by a formula:

un=f(n)

or by arecurrence relation(difference equation), such as:

un=un− 1 +un− 2 u 1 =a, u 2 =b

Problem 14.8

Write down an expression or a recurrence relation for thenth termunin

each of the following cases:

(i) 1, 4, 9, 16, 25,... (ii) 1,

1

2!

,

1

3!

,

1

4!

,

1

5!

,....

(iii) 1, 4, 7, 10,... (iv) 2, 4, 10, 24, 58...

This is really a matter of educated guesswork – find a result that fits the first few terms
of the sequence, check that it holds for the other terms and assume that it holds in general
for all terms, even those that are not explicitly written down.

(i) We note that the terms can be written as 1^2 , 22 , 32 , 42 , 52 ,...which

suggests that in general we can take

un=n^2

(ii) The same idea suggestsun=

1

n!

in this case.