8.3 The Integral and Comparison Tests; Estimating Sums of 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
    
Code for the Mathematica procedures AreaR and AreaL is modified slightly from programs developed in
    Finch and Lehmann, "Exploring Calculus with
Mathematica," Addison-Wesley, 1992

AreaR and  AreaL -- contains Mathematica code only, not worked exercises

The Integral Test for Convergence of Infinite Series

Software that we used to illustrate Riemann Sums in an earlier chapter can be reused in the present context to illustrate the relationship between the convergence of certain integrals and the convergence of related infinite series.

In[187]:=

<<Graphics`Colors`

In[188]:=

Clear[f, x, a, b, k, n] ; <br /> f[x_] := 1/x^2 ; <br /> a = 1 ;       ... ;    (* n = number of rectangles *)AreaRVerbose[f, a, b, n] ;

[Graphics:HTMLFiles/8.3_sumOfSeries_4.gif]

RowBox[{The total area of the illustrated rectangles is , , 0.491389}]

The area of the illustrated rectangles is the sum of the first 5 terms of an infinite series.

In[191]:=

Underoverscript[∑, k = 2, arg3] 1/k^2 %//N

Out[191]=

1769/3600

Out[192]=

0.491389

The figure illustrates that this sum of five terms must be less than the value of an associated integral.

In[193]:=

∫_1^61/x^2x %//N

Out[193]=

5/6

Out[194]=

0.833333

Similarly, the sum of the infinite series must be less than the value of a certain improper integral.

In[195]:=

Underoverscript[∑, k = 2, arg3] 1/k^2 %//N

Out[195]=

1/6 (-6 + π^2)

Out[196]=

0.644934

In[197]:=

∫_1^∞1/x^2x

Out[197]=

1

This sort of comparison is the basis for the Integral Test for convergence of infinite series.


Created by Mathematica  (May 5, 2004)