Understanding Engineering Mathematics

l

n

Figure 14.9A sequence approaching a limitl.

If a sequence has a finite limit, we say itconverges, if not, itdiverges. Graphi-
cally, a sequence converges if the points plotted for values ofnapproach a definite line
(Figure 14.9).
Note that the behaviour of the sequence depends not at all on the first terms of the
sequence, but rather on the ‘infinite tail’ – i.e. on the last infinity of terms!

Exercise on 14.7

Investigate the convergence of the sequences

un= (i) (− 1 )n (ii) 2n (iii)

(

1

3

)n

(iv) −

1

n

(v)

1

rn

|r|>1(vi)

1

rn

|r|< 1

Answer

(i) D (‘oscillating’) (ii) D (iii) C (iv) C (v) C (vi) D

14.8 Iteration

One of the most important practical applications of the theory of sequences is in the discus-
sion ofiterative methodsused for example in numerical methods for solving equations. In
an iterative method for solving an equation we assume an initial guess and use it in some
rearrangement of the equation to calculate a (hopefully) improved approximation to the
solution. We will describe two methods for this, but mainly as examples of convergence
of sequences.
One way of finding an approximate solution to the equation

f(x)= 0

is to write it in the form

f(x)≡F(x)−x= 0

and solve this equation by iteration:

xn+ 1 =F(xn)