194 Chapter 7Sequences and series
limit is not finite or not unique the sequence is divergent. For example, the arithmetic
progression (7.1) is necessarily divergent for all values of d:
The geometric progression (7.3) is interesting because it is convergent for some
values of xand divergent for others. It shows the six possible types of behaviour,
characteristic of many sequences, illustrated in Figure 7.1.
The examples shown are (fora 1 = 11 ):
(a) x 1 > 11 1,2,4,8,=
(b)x 1 = 11 1,1,1,1,=
(c) 0 1 < 1 x 1 < 11
(d)− 11 < 1 x 1 < 10
(e) x 1 = 1 −11,−1, 1,−1,=
(f) x 1 < 1 −11,−2, 4,−8,=
The sequence is convergent only forx 1 = 1 + 1 and|x| 1 < 11 , types (b), (c), and (d).
The behaviour of a sequence in the limit does not necessarily depend on the
behaviour of a finite part of the sequence. An important example is the sequence of
the terms {x
r2 r!} in the expansion of the exponential function (Section 3.6). This
sequence has limit zero for all values of x; thus, the ratio of consecutive terms is
but the terms increase in magnitude whenr 1 + 111 < 1 |x|, decrease whenr 1 + 111 > 1 |x|. For
example,
e
323451
3
1
3
2
3
3
3
4
3
5
=++++++=++ + +1 3 45 45
!!!!!
.. 3 3 375 2 025..++
x
r
x
r
x
r
r
rr+
=
→→
11
10
()!!
as ∞∞ for all x
1
1
2
1
4
1
8
,− , ,− ,...
1
1
2
1
4
1
8
,,,,...
lim ( )
r
adr
→
+−
=±
∞
∞1
.......................................................................................................................................................................................................................................................................................10
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u
r0
(a)
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(b)
.......................................................................................................................................................................................................................................................................................1
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(c)
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(d)
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(e)
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r
(f)
Figure 7.1