8.8 Binomial Series

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
    Selwyn Hollis, "CalcLabs with
Mathematica" for Stewart's "Single Variable Calculus, Concepts and Contexts, Second Edition," Brooks/Cole, 2001

A Specific Binomial Series

cf. Stewart, Section 8.8, Example 1, page 624

Let's start with a specific function whose Taylor series is a binomial series.

In[427]:=

<<Graphics`Colors`

In[428]:=

Clear[f, x, fSeries] ;  f[x_] := 1/(1 + x)^2                                   ...                                                                                         (1\ + \x)

[Graphics:HTMLFiles/8.8_binomialSeries_3.gif]

The power series representation of this function is a particularly simple binomial series.

In[431]:=

Series[f[x], {x, 0, 10}] fSeries[x_] = Normal[%]

Out[431]=

1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - 6 x^5 + 7 x^6 - 8 x^7 + 9 x^8 - 10 x^9 + 11 x^10 + O[x]^11

Out[432]=

1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - 6 x^5 + 7 x^6 - 8 x^7 + 9 x^8 - 10 x^9 + 11 x^10

The function f has a singularity at x = -1, but its approximating polynomial has no singularities whatsoever. So just how similar are these two functions?

In[433]:=

Plot[{f[x], fSeries[x]}, {x, -2, 2}, PlotRange {-1, 50}, PlotLabel-> ... ;AxesLabel {"x", None}, PlotStyle {DarkTurquoise, TerreVerte}] ;

[Graphics:HTMLFiles/8.8_binomialSeries_8.gif]

This particular partial sum of the power series representation of f (a binomial series) seems to be doing a very good job of tracking f over much of the interval (-1,1), but it is hopeless outside of this interval. The theory developed in Stewart, Section 8.8, explains this behaviour. The power series representation is a binomial series expansion about x = 0, and it has radius of convergence R = 1.

Another Binomial Series

cf. Stewart, Section 8.8, Example 2, page 624

Let's start with another specific function whose Taylor series is a binomial series.

In[434]:=

Clear[f, x, fSeries] ;  f[x_] := 1/(4 - x)^(1/2)                               ... 1;PlotStyleDarkViolet] ;                                                      Sqrt[4 - x]

[Graphics:HTMLFiles/8.8_binomialSeries_10.gif]

The power series representation of this function is a binomial series.

In[437]:=

Series[f[x], {x, 0, 10}] fSeries[x_] = Normal[%]

Out[437]=

1/2 + x/16 + (3 x^2)/256 + (5 x^3)/2048 + (35 x^4)/65536 + (63 x^5)/524288 + (231 x^6)/8388608 ... )/67108864 + (6435 x^8)/4294967296 + (12155 x^9)/34359738368 + (46189 x^10)/549755813888 + O[x]^11

Out[438]=

1/2 + x/16 + (3 x^2)/256 + (5 x^3)/2048 + (35 x^4)/65536 + (63 x^5)/524288 + (231 x^6)/8388608 ... + (429 x^7)/67108864 + (6435 x^8)/4294967296 + (12155 x^9)/34359738368 + (46189 x^10)/549755813888

The function f has a singularity at x = 4, but its approximating polynomial has no singularities whatsoever. So just how similar are these two functions?

In[439]:=

Plot[{f[x], fSeries[x]}, {x, -2, 4}, PlotLabel->"The Function f and an Approxi ... #62371;AxesLabel {"x", None}, PlotStyle {DarkViolet, Peacock}] ;

[Graphics:HTMLFiles/8.8_binomialSeries_15.gif]

This particular partial sum of the power series representation of f (a binomial series) seems to be doing a very good job of tracking f over much of the interval (-4,4), but it is hopeless outside of this interval. In fact, f is not even defined for x > 4, but the Taylor Polynomial is defined for all x. The theory developed in Stewart, Section 8.8, explains this behaviour. The power series representation is a binomial series expansion about x = 0, and it has radius of convergence R = 4.


Created by Mathematica  (May 5, 2004)