8.4 Other Convergence Tests

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

Alternating Series

In an alternating series the signs of the terms alternate.

In[208]:=

<<Graphics`Colors`

In[209]:=

Clear[k, n] ;  n = 10 ;  seq = Table[(-1)^k/k, {k, n}] <br /> Underoverscript[∑, k = 1, arg3] (-1)^k/k

Out[211]=

{-1, 1/2, -1/3, 1/4, -1/5, 1/6, -1/7, 1/8, -1/9, 1/10}

Out[212]=

-1627/2520

The three requirements of the alternating series test are satisfied for this series, so it converges, in spite of its resemblence to the divergent harmonic series.

In[213]:=

Underoverscript[∑, k = 1, arg3] (-1)^k/k

Out[213]=

-Log[2]

This plot suggests that convergence, and even indicates why the criteria of the alternating series test will guarantee convergence of an alternating series. The oscillations about the limit point must tend to zero.

In[214]:=

Clear[partialSum, partialSums, n, k] ;  partialSum[n_] :=   Underoverscript[ ... p;       , AxesLabel {"n", None}}], ]}], ;}]

[Graphics:HTMLFiles/8.4_otherTests_8.gif]


Created by Mathematica  (May 5, 2004)