8.5 Power 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 Power Series

cf. Stewart, Section 8.5, Example 5, page 604

Let's find the radius of convergence and interval of convergence of a specific power series.

In[244]:=

<<Graphics`Colors`

In[245]:=

Clear[a, n, seq, ratio] ;  a[n_] := n x^n/3^(n + 1) ;  seq = Table[a[n], {n, 10}]

Out[247]=

{x/9, (2 x^2)/27, x^3/27, (4 x^4)/243, (5 x^5)/729, (2 x^6)/729, (7 x^7)/6561, (8 x^8)/19683, x^9/6561, (10 x^10)/177147}

The Ratio Test is the appropriate instrument for determining the radius of convergence of the power series with nth term a[n].

In[248]:=

ratio[n] = Abs[a[n + 1]/a[n]]

Out[248]=

1/3 Abs[((1 + n) x)/n]

The ratio test takes the limit of this ratio as n → ∞, and compares the result to the number 1.

In[249]:=

Limit[ratio[n], n∞]

Out[249]=

Abs[x]/3

Since x is real, we require |x|/3 < 1, or |x| < 3. The radius of convergence is 3.
Let's check for convergence at the endpoints of the interval of convergence.
When x = 3, a[n] = n/3, and the series diverges because its nth term does not approach zero.
When x = -3, a[n] = (-1)^n n/3, and once again the series diverges since its nth term does not approach zero.
We conclude that our series converges on the open interval (-3,3) and diverges elsewhere.

In[250]:=

Clear[f, x, m]  f[x_, m_] := Underoverscript[∑, n = 1, arg3] n x^n/3^(n + 1)  ... 1;PlotRange {0, 10}, PlotStyle {Tomato, MediumOrchid, MediumTurquoise}] ;

[Graphics:HTMLFiles/8.5_powerSeries_10.gif]

For some reason, Mathematica gets confused when trying to sum this infinite series for negative values of x, so we will plot its graph only for positive x's.

In[253]:=

Clear[f, x, m]  f[x_] := Underoverscript[∑, n = 1, arg3] n x^n/3^(n + 1)  ... ot[f[x], {x, 0, 3}, PlotStyleOliveDrab, PlotLabel->"f(x)"] ;

[Graphics:HTMLFiles/8.5_powerSeries_12.gif]

Restricting the range of f(x) gives a little more information on its behaviour near the x-axis.

In[256]:=

Plot[f[x], {x, 0, 3}, PlotStyleOliveDrab, PlotRange {0, 10}, PlotLabel->"f[x]"] ;

[Graphics:HTMLFiles/8.5_powerSeries_14.gif]


Created by Mathematica  (May 5, 2004)