Graphics Reference
In-Depth Information
12
Sequences of functions
For this chapter, more than any other, the availability of a
computer-assisted graph drawing package is essential, and the
preliminary exercise (i)
—
(x) is dependent on this
Concurrent reading:
Burkill and Burkill ch. 5, Spivak ch. 23.
Further reading:
Rudin ch. 7.
Sequences of functions are often considered in a second course of
analysis. The diMculties encountered are not like those in earlier
analysis since the apparatus of limits has already been developed.
DiMculties arise from having two variables of unsymmetric status and
having to think of keeping one constant and letting the other vary
according to context. Before starting the questions in this chapter it is
necessary to have some intuitive feel for the subject matter, and to have
observed, in graphical form, how a sequence of functions may behave.
To do this, use graph drawing facilities on a computer to illustrate the
functions
f
for
n
1, 2, 5 and 10, where
(i)
f
(
x
)
x
1/
n
,
(ii)
f
(
x
)
x
/
n
,
nx
x
n
for
x
R
,
(iii)
f
(
x
)
1
x
n
(vi)
f
(
x
)
,
1
1
n
x
(v)
f
(
x
)
,
nx
1
n
x
(vi)
f
(
x
)
,
x
(vii)
f
(
x
)
,
n
x
1
