In article
<aa9545df-690b-4565-8497-ef2e97c78da5@[EMAIL PROTECTED]
>,
Vladimir Bondarenko <vb@[EMAIL PROTECTED]
> wrote:
>An interesting catch of yours!
>Indeed,
>NSum[Cos[n]^2/n^2, {n,1,Infinity}, Method -> "AlternatingSigns"]
>NSum[Cos[n]^2/n^2, {n,1,Infinity}, Method -> "WynnEpsilon"]
>NSum[Cos[n]^2/n^2, {n,1,Infinity}, Method -> "EulerMaclaurin"]
>0.543756
>0.553446
>0.574138
>So Mathematica 6.0.2 *had* a clear chance to produce a
>nice answer (0.574138)... but it missed it...
>Maybe someone from Wolfram Research could enlighten us
>the customers why such a misfire happened?
>On Mar 25, 12:55=A0am, SzH <szhor...@[EMAIL PROTECTED]
> wrote:
>> On Mar 25, 5:28 am, Vladimir Bondarenko <v...@[EMAIL PROTECTED]
> wrote:
>> > NSum[(Cos[n]/n)^1, {n, 1, Infinity}] =A0 =A0 slow convergence
>> > NSum[(Cos[n]/n)^2, {n, 1, Infinity}] =A0 =A0 fast convergence
>> > 0.0420195 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0RIGHT
>> > 0.572578 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 WRONG
>> > Any comments?
>> Interestingly, the setting Method -> "EulerMaclaurin" gives a much
>> more precise answer (0.574138), however, no explicit setting for
>> Method can reproduce the answer that we get with Method -> Automatic.
All of this is not surprising. "Automatic" manages to
find an analytic function it knows how to compute.
One has to use results from analysis to even prove the
first series converges. However, one questions how many
terms were used by the various methods, or how much
accuracy was lost by what are essentially rounding methods.
Rubin's general comment: Do analysis before numerical
analysis. "Purely" numerical methods usually do not
work well.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@[EMAIL PROTECTED]
Phone: (765)494-6054 FAX: (765)494-0558
|
