5128_Ch08_434-471

438 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
Then a 1 • r 6 implies that a 1 3. The sequence is defined explicitly by
a n (3)(2) n1 (1) n (3)(2 n1 ).
Now try Exercise 21.
Graphing a Sequence
As with other kinds of functions, it helps to represent a sequence geometrically with its
graph. One way to produce a graph of a sequence on a graphing calculator is to use parametric
mode, as shown in Example 6.
EXAMPLE 6 Graphing a Sequence Using Parametric Mode
Draw a graph of the sequence {a n } with a n (1) n n 1
, n 1, 2, … .
n
[0, 20] by [–2, 2]
Figure 8.1 The sequence of Example 6.
SOLUTION
Let X 1T T, Y 1T (1) T T 1
, and graph in dot mode. Set Tmin 1, Tmax 20,
T
and Tstep 1. Even through the domain of the sequence is all positive integers, we are
required to choose a value for Tmax to use parametric graphing mode. Finally, we choose
Xmin 0, Xmax 20, Xscl 2, Ymin 2, Ymax 2, Yscl 1, and draw the graph
(Figure 8.1). Now try Exercise 23.
Some graphing calculators have a built-in sequence graphing mode that makes it easy
to graph sequences defined recursively. The function names used in this mode are u, v, and
w. We will use this procedure to graph the sequence of Example 7.
EXAMPLE 7
Graphing a Sequence Using Sequence Graphing Mode
Graph the sequence defined recursively by
SOLUTION
b 1 4
b n b n1 2 for all n 2.
We set the calculator in Sequence graphing mode and dot mode (Figure 8.2a). Replace b n
by u(n). Then select nMin 1, u(n) u(n 1) 2, and u(nMin) {4} (Figure 8.2b).
u=u(n–1)+2
n=1
X=1 Y=4
Normal Sci Eng
Float 0123456789
Radian Degree
Func Par Pol Seq
Connected Dot
Sequential Simul
Real a+bi re^i
Full Horiz G–T
(a)
Plot1 Plot2 Plot3
nMin=1
u(n)=u(n–1)+2
u(nMin)= 4}
v(n)=
v(nMin)=
w(n)=
w(nMin)=
(b)
[0, 10] by [–5, 25] Figure 8.2 (a) Setting sequence mode and dot mode on the calculator, and (b) entering
the sequence of Example 7 in the calculator.
Figure 8.3 The graph of the sequence
of Example 7. The TRACE feature shows
the coordinates of the first point (1, 4) of
the sequence,
b 1 4, b n b n1 2, n 2.
Then set nMin 1, nMax 10, PlotStart 1, PlotStep 1, and graph in the [0, 10] by
[5, 25] viewing window (Figure 8.3). We have also activated Trace in Figure 8.3.
Now try Exercise 27.