Explore

**Wolfram Demonstrations on the subject** - a lot of free code to download and nice looking applications. For example:

**Taylor Polynomials by Harry Calkins** Manipulate[

Column[{Style[Row[{"series degree = ", deg}], "Label", 12],

Plot[Evaluate@{func /. x -> t,

MakeSeriesFunction[func, x, deg, pt]}, {t, 0, 2 Pi},

PlotRange -> {{0, 2 Pi}, {-2, 2}},

PlotStyle -> {{GrayLevel[.25, .85]}, {RGBColor[0, 0, .9, .8]}},

Filling -> {1 -> {{2}, Directive[{Opacity[.1], Green}]}},

Epilog -> {Red, PointSize[.02], Point[{pt, func /. x -> pt}],

Blue, Point[{pt, 0}], Dashing[{.01}],

Line[{{pt, func /. x -> pt}, {pt, 0}}]},

ImageSize -> {500, 350}]}, Center], {{deg, 10, "series degree"},

1, 24, 1}, {{pt, 0, "expansion point"}, 0,

2 Pi}, {{func, Sin[x],

"function"}, {Sin[x] -> TraditionalForm[Sin[x]],

Cos[x*2]/(x + 1) -> TraditionalForm[Cos[x*2]/(x + 1)],

1/(1 + x) -> TraditionalForm[1/(1 + x)],

E^x/100 -> TraditionalForm[E^x/100],

2 E^-(-3 + x)^2 -> TraditionalForm[2 E^-(-3 + x)^2]}},

SaveDefinitions -> True,

Initialization :> {MakeSeriesFunction[fn_, var_, deg_, pt_] :=

Module[{wrkfn},

wrkfn[tt_] =

If[Head[fn] === Symbol && fn =!= var, fn[tt], fn /. var :> tt];

Normal@Series[wrkfn[t], {t, pt, deg}]

]}]