## Series Applet

Series illustrates the concept of a series and has options to explore various convergence tests. In its basic form it shows a graph of the successive partial sums of a series as discrete dots. It plots the terms of the sequence being summed as rectangles, so that the partial sum for any given number of terms is just the sum of the areas of the rectangles to the left. It also displays a three-column table, where the first column provides different values for n and the second two columns present values that depend on the specific type of convergence test being illustrated by the applet.

The applet tag that creates the applet is as follows:

<applet code="net.sourceforge.webcompmath.calculus.Series" codebase="../../assets"

archive="webcompmath.jar,calculus.jar" height="400" width="640">

<param name="UseLoadButton" value="no" />

<param name="Col1Title" value="<html>s<sub>n</sub></html>" />

<param name="Col1Def" value="a(n)" />

<param name="Col2Title" value="<html>S<sub>n</sub></html>" />

<param name="Col2Def" value="as(n)" />

<param name="Example1" value="1. Arithmetic Series;s1 + d(n-1),s;;10;0,10,0,120;1,s1,0,5;2,d,0,5;10,50,100,500,1000,5000;false" />

<param name="Example2" value="2. Geometric Series;s1*r^(n-1),s;;10;0,10,-1100,1100;1,s1,0,5;2,r;10,50,100,500,1000,5000;false" />

<param name="Example3" value="3. A Divergent Series;(n+1)/n,s;;10;0,10,0,13;;;10,50,100,500,1000,5000;false" />

<param name="Example4" value="4. Harmonic Series;1/n,s;;10;0,10,0,4;;;10,50,100,500,1000,5000;false" />

<param name="Example5" value="5. P-Series;1/n^p,s;;10;0,10,0,4;1,p;;10,50,100,500,1000,5000;false" />

</applet>

This applet takes the following parameters:

Param NameTypeDefaultDescriptionExample1, ...String N/A Each example is a list of strings separated by semicolons. The strings, in order, are:

1. The name of the example, displayed in the example menu.

2. The expression for a(n), the definition of a sequence term. If empty (i.e., if ""), then the a input box and label are hidden. Can also be following by an optional command and string to provide the name for the sequence term. The default name is "a", but can be set to something else (like "s").

3. The expression for b(n), the definition of a second sequences terms. If empty (i.e., if ""), then the b input box and label are hidden. Can also be following by an optional command and string to provide the name for the sequence term. The default name is "b".

4. The value for nmax, the number of points to plot.

5. The xmin, xmax, ymin, and ymax for the graph, as a comma-separated list of numbers.

6. Defines a parameter to be added at the bottom of the applet. The definition is a comma-separated list; if the list is "", then no parameter is displayed. The first element is the initial value of the parameter. The second element is optional and gives the name of the parameter (default is "p1"). The optional third and fourth elements give the minimum and maximum range values for the parameter slider (if not provided, they default to -5 and 5). For example, "2,d,0,5" means "create a parameter called "d" with an initial value of 2, and limit the parameter's slider to value from 0 through 5." The name of the parameter can then be used in expressions that define the a(n) and b(n) sequences. See the "Arithmetic Series" example in the applet code above.

7. Defines a second parameter to be added at the bottom of the applet. The syntax is the same as above, except that the default name is "p2".

ShowSecondyes/no no If "yes", then a second series is shown, with its own definition input box, graphed points, and graphed rectangles. If "no" only one series is graphed. ShowGraphyes/no no If "yes", then a continuous graph is shown for the sequence terms. If "no" then no continuous graph is shown. Mostly used for illustrating the integral test. Col1Title, Col2Titlestring n/a The titles of the second and third table columns (the first column is column 0 and is always "n"). These can be either simple text or html. To use html, however, you can't just use < and > directly, since the web page in which the applet is embedded will think these characters are meant for it. Instead, you need to code your html using <and>. For example, to make a column 1 heading ofnwould would just use<param name="Col1Title" value="n" />. But if you wanted to make this bold using html, you cannot say<param name="Col1Title" value="<html><b>n</b></html> />because the embedded < and > cause problems. Instead, use<param name="Col1Title" value="<html><b>n</b></html> />. This is much easier to do if you create your web pages with an html editor like Dreamweaver, which allows you to just type < and > in design view and it takes care of converting these to<and>in the actual html code.Col1Def, Col2Defstring n/a The expressions for the values computed in the second and third columns (the first column is column 0 and is always "n"). The following functions and variables are of interest:

n: the index of the term in the series

a(n): the a sequence

b(n): the b sequence (only useful if ShowSecond is set to "yes"

as(n): the a series partial sums

bs(n): the b series partial sums

nmax: the upper limit for the partial sums to be graphed.