Re: strange errors for numerical evaluation of an infinite sum
- To: mathgroup at smc.vnet.net
- Subject: [mg58364] Re: [mg58322] strange errors for numerical evaluation of an infinite sum
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Tue, 28 Jun 2005 21:56:58 -0400 (EDT)
- References: <200506280913.FAA05127@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Roger Bagula wrote:
> The Sum is:
> aa = Sum[1/(Prime[n + 1] - Prime[n])^PrimePi[n], {n, 1, Infinity}]
> The numerical evaluation is:
> N[aa]
>
> The error is:
> \!\(\*
> RowBox[{\(Prime::"intpp"\), \(\(:\)\(\ \)\), "\<\"Positive integer
> argument
> expected in \\!\\(
> Prime[16.`]\\). \\!\\(\\*ButtonBox[\\\"More?\\\", \
> ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
> ButtonData:>\\\"Prime::intpp\\\"]\\)\"\>"}]\)
>
> and:
>
> \!\(\*
> RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output
> of \\!\\(Prime :: \\\"intpp\\\"\\) will be suppressed during this \
> calculation. \\!\\(\\*ButtonBox[\\\"More?\\\", \
> ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
> ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]\)
>
> I got a slightly better answer by extending: $MaxExtraPrescision
> but I'm only getting a number to 19 places , it appears.
>
> Any help will be appreciated. It seem to be some kind of frequency
> number associated with the Primes.
> I don't know if it is a new constant or not.
>
> Respectfully,
> Roger L. Bagula
Sum it to a finite bound.
In[1]:= aa[j_] := Sum[1/(Prime[n + 1] - Prime[n])^PrimePi[n], {n, j}]
In[2]:= N[aa[10000]]
Out[2]= 2.1168
It is not too hard to bound the error. The base of the denominator is
always at least 2 (and at most Prime[n] which is again easy to bound),
and PrimePi[n] has asymptotic behavior that is easy to estimate
sufficiently closely for this purpose.
Daniel Lichtblau
Wolfram Research
- References:
- strange errors for numerical evaluation of an infinite sum
- From: Roger Bagula <rlbagulatftn@yahoo.com>
- strange errors for numerical evaluation of an infinite sum