Series Approximations

SetOptions[Manipulate,BaseStyle->{FontSize->20}];
SetOptions[Manipulator,Appearance->{"Labeled"},BaseStyle->{FontSize->16}];
SetOptions[Plot,AxesLabel->Automatic,PlotStyle->Thick,ImageSize->700,BaseStyle->{FontSize->16}];

The purpose of this activity is to explore the behavior of series approximations (of various orders and at different points) relative to the original function.

We begin by choosing a function to approximate via series expansion.

f[z_]=Sin[z];

Before proceeding, we specify domain and range values for the plot window. These can be adjusted later, if necessary.

Note that these values have no effect on the calculation itself.

PlotRangeValue=4;
PlotDomainValue = 5;

Now we plot the function (blue) along with its expansion (orange). We have two input sliders to play with: ExpansionOrder specifies the expansion’s order, and ExpansionPoint specifies the z-coordinate around which the expansion is performed (black dot).

Note that this code block does not need to be modified (but feel free to play around with it!).

SetOptions[Plot,PlotRange->{-PlotRangeValue,PlotRangeValue}];
Manipulate[
SeriesExpansion[z_]=Series[f[z],{z,ExpansionPoint,ExpansionOrder}]//Normal;
Grid[{
{Plot[{f[z],SeriesExpansion[z]},
{z,-PlotDomainValue,PlotDomainValue},Epilog->{Black,PointSize@Large,Point[{ExpansionPoint,f[ExpansionPoint]}]}]},
{StringForm["Expansion: ",ExpansionOrder][SeriesExpansion[z]]}},
Alignment->Left],
{ExpansionOrder,0,5,1},
{{ExpansionPoint,0},-PlotDomainValue,PlotDomainValue,.1},TrackedSymbols:>{ExpansionPoint,ExpansionOrder}]

Where does the expansion agree well with the original function?

What minimum expansion order is necessary to well approximate the original function? Does your response depend on the location of the expansion? Is this true for all input functions?

By Corinne Manogue and Ian Founds
Copyright 2000 Corinne Manogue