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.
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