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>