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[MUG] conditional convergence error
| [MUG] conditional convergence error |
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Author: Barsuhn
Posted: 23/03/2000 17:16:22 GMT
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>> From: Barsuhn "barsuhn"
Ladies and Gentlemen,
one mathematical topic often surprising for beginners is that the sum of
a series may change upon a rearrangement of its terms. For this purpose
you may just look at the sequence of partial sums using a simple program
e.g. in BASIC. However, it would also be nice to use Maple as a "black
box expert" on these problems. Maple succeeds with the simple Leibniz
series
1 - 1/2 + 1/3 - 1/4 + - etc. = ln2 and e.g. its "++-" rearrangement
1+1/3 -1/2 +1/5+1/7 -1/4 etc =3/2 ln2.
But if we replace the denominators by its square roots, Maple correctly
finds the sum of the "original" series. Unfortunately, the result is
wrong for the "++-"-rearrangement. This rearrangement diverges to
+infinity. Maple V up to r5 yields the sum of the original series, as if
the rearrangement would have no effect on the sum. Maple 6 (at least the
trial version) even finds a negative sum ! As the reason for these
failures is hidden deeply in the evalf-C-code, there seems to be no
chance for a cure.
For details, please have a look on the following part of a worksheet:
The following series No. 1 is (conditionally) convergent. Maple finds
its sum
> restart;sum((-1)^(n-1)*1/sqrt(n),n=1..infinity); # No. 1
> evalf(%);
>
The following series No. 2 is a rearrangement of the series No. 1. The
sum of this rearrangement is + infinity. However, Maple gives the sum of
the original series (Maple V up to r5) and thus a wrong result.
> umord:=sum(1/sqrt(4*n-3)+1/sqrt(4*n-1)-1/sqrt(2*n),n=1..infinity); #
No. 2
> evalf(umord); # The result given is wrong !!!
By looking at the sequence of the terms, we see that "umord" is indeed a
rearrangement of the first series.
> seq([1/sqrt(4*n-3),+1/sqrt(4*n-1),-1/sqrt(2*n)],n=1..10);
The asymptotic expansion of the terms in parentheses show that they
approach zero not faster than const/sqrt(n). As the series 1 +
1/sqrt(2)+ 1/sqrt/3) + .. . .. is divergent, series No. 2 is divergent
to + infinity.
> asympt(1/sqrt(4*n-3)+1/sqrt(4*n-1)-1/sqrt(2*n),n);
All the best Jurgen
--
-------------------
Prof. Dr. Jurgen Barsuhn
Fachhochschule Bielefeld
University of Applied Sciences
Fachbereich Elektrotechnik und Informationstechnik
D-33511 Bielefeld
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