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Fig. 2-1

6 9 5 7 8 5 0 0 0 0

4 9 3 2 4 2 3 0 0 0

8 0 3 5 0 0 5 3 0 0

Using (1) and (2) in (3), we see that jup uq j > b a =3 for in nitely many values of p and q, thus contradicting the hypothesis that jup uq j < for p; q > N and any > 0. Hence, there is only one limit point and the theorem is proved.

- 148 -

9 5 3 3 9 5 5 6 3 0

1 8 5 4 0 0 7 8 7 4

money is borrowed and r is the annual rate of interest. Construct a table showing the monthly payments on a 30-year mortgage of $250,000 at rates of 6% to 8% in increments of 25%.

INFINITE SERIES 2.25. Prove that the in nite series (sometimes called the geometric series) a ar ar2 (a) converges to a= 1 r if jrj < 1,

12n

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list = Table[{PaddedForm[r, {4, 4}], PaddedForm[payment, {6, 2}]}, {r, .06, .08, .0025}]; TableForm[list, TableHeadings {None, {" rate", " payment"}}] rate 0.0600 0.0625 0.0650 0.0675 0.0700 0.0725 0.0750 0.0775 0.0800 payment 1498.88 1539.29 1580.17 1621.50 1663.26 1705.44 1748.04 1791.03 1834.41

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Anyone who has ever tried to plot a graph using one of the standard programming languages will appreciate the ease with which graphs can be produced in Mathematica. In many instances, only one simple instruction is all that is needed to produce a pictorial representation of a function or a more general relationship between two variables. Although Mathematica s defaults work well in most instances, there are many options available to control subtleties. We shall describe the more common ones in this section and present a variety of examples that illustrate the ease with which graphs may be constructed. The basic command for drawing the graph of a function is Plot. Although x is used as the independent variable in the following description, any symbol may be used in its place.

Let Then Subtract,

Plot[f[x], {x, xmin, xmax} plots a two-dimensional graph of the function f(x) on the interval xmin x xmax.

Plot[x2, {x, 3, 3}]

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n!1 n!1

Plot[{f[x], g[x]}, {x, xmin, xmax}] plots the graphs of f(x) and g(x) from xmin to xmax on the same set of axes. This command can be generalized in a natural way to plot three or more functions.

Plot[{x2, 9 x2}, {x, 3, 3}]

When plotting points over a specified interval, Mathematica makes a decision on the range of points to plot in order to produce a pleasing graph. PlotRange is an option that allows the user to override Mathematica s default.

(b) If jrj > 1, lim Sn does not exist (see Problem 44).

PlotRange Automatic is Mathematica s default. Any points whose vertical coordinates appear to be too large (e.g., outliers) are omitted from the graph. PlotRange All forces Mathematica to plot all points. PlotRange {ymin, ymax} plots only those points whose vertical coordinates fall between ymin and ymax. PlotRange {{ xmin, xmax } , { ymin, ymax }} plots those points whose horizontal coordinates fall between xmin and xmax and whose vertical coordinates fall between ymin and ymax.

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