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Topic: SumConvergence still imperfect
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Andrzej Kozlowski

Posts: 2,112
Registered: 1/29/05
SumConvergence still imperfect
Posted: Jan 22, 2012 7:19 AM
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The function SumConvergence in Mathematica 8 performs much better than
in 7 (where it first appeared) but it is still not perfect. Consider
this example:

SumConvergence[(-1)^((1/2)*n*(n + 1))/Log[n], n]

False

Well, the sum is actually convergent by the Dirichlet test (the sequence
1/Log[n] is monotonic with limit 0 and the sequence {1,1,-1,-1,1,1,...}
has bounded partial sums.

Wolfram Alpha also gives the wrong answer.

However, in the next case, where exactly the same argument can be used,
SumConvergence gets it right:

SumConvergence[(-1)^(1/2 n (n + 1))/n, n]

True


However Wolfram Alpha still can't do this. Asked to evaluate

Sum[(-1)^(n (n + 1)/2)/n, {n, 1, Infinity}]

it first given an inconclusive answer (claiming to have run out of
time). Given more time it returns an approximate answer, which is the
same as the one given by NSum

NSum[(-1)^(n (n + 1)/2)/n, {n, 1, Infinity}]

During evaluation of In[105]:= NSum::emcon: Euler-Maclaurin sum failed
to converge to requested error tolerance. >>

-1.10245 - 0.268775 I

which is obviously wrong. Unfortunately Wolfram Alpha does not issue the
warning that the answer is essentially meaningless so some poor soul
might really believe that the limit of a series of real numbers could be
imaginary.

One can get a much better approximate answer using Sum with a large
number of terms, e.g.

N[Sum[(-1)^(n (n + 1)/2)/n, {n, 1, 10^5}], 5]

-1.1320

Andrzej Kozlowski




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