Re: another problem with Infinite Products

*To*: mathgroup at smc.vnet.net*Subject*: [mg65522] Re: [mg65481] another problem with Infinite Products*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Thu, 6 Apr 2006 06:52:12 -0400 (EDT)*References*: <200604051055.GAA21639@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Roger Bagula wrote: > Gary Adamson a long time ago came up with a sort of ultra-zeta sum: > A=Sum[1/n^n,{n,1,Infinity}]=1.291.. > That is very close to 2- Sqrt[2]... > One wonders if the Product: > Product[1/(1-1/Prime[n]^Prime[n]),{n,1,Infinity}]-->A > or is it? > Product[1/(1-1/Prime[n]^n),{n,1,Infinity}]-->A > It is clear that both these products should converge > and I think that the second should be A on reconsideration. > But Mathematica disagrees: > > A = Sum[1/n^n, {n, 1, Infinity}] > N[%] > 1.2912859970626636` > Product[1/(1 - 1/Prime[n]^Prime[n]), {n, 1, Infinity}] > N[%] > 1.3850602852044895` > Product[1/(1-1/Prime[n]^n),{n,1,Infinity}] > N[%] > 2.2691047868959395` I am not sure what is meant in the subject header by the phrase "problem with Infinite Products". If it refers to this being "another example" that involves infinite products, well and good. If it means "problem" as in "bug", well, that would be incorrect. I should mention, in case it is not already clear, that these products do not have known closed forms in terms of defined functions. Hence they return unevaluated. Application of N in Mathematica will invoke NValues that evaluate them as truncations, possibly with some form of extrapolation. As will become clear below, at least two of the numeric evaluations are quite reasonable. As for the specifics of these examples, it is clear that Product[1/(1 - 1/Prime[n]^Prime[n]), {n, 1, Infinity}] will be smaller than Product[1/(1-1/Prime[n]^n),{n,1,Infinity}] To see that the former is larger than n2nrecipsum = Sum[1/n^n, {n, 1, Infinity}] is fairly straightforward and mostly elementary math. Details are provided below. We hereafter take for granted that each product converges though this too is not difficult to prove. First we define for notation purposes pnpn[n_] := 1/Prime[n]^Prime[n] Then our product may be written pp = Product[1/(1-pnpn[n]),{n,1,Infinity}] Since all factors are positive we know it is the exponential of its logarithm, so we work with logpp = Sum[Log[1/(1-pnpn[n])], {n,1,Infinity}] Since pnpn[n]<1 we can expand each log as an infinite sum about zero, obtaining logpp = Sum[Sum[pnpn[n]^j/j, {j,1,Infinity}], {n,1,Infinity}] Here we require some post-calculus math: as the stars are properly aligned we can switch the order of summation, to obtain logpp = Sum[Sum[pnpn[n]^j/j, {n,1,Infinity}], {j,1,Infinity}] We still cannot sum this explicitly, but we can minorize it by truncation, and the truncated versions can be summed. logppreduced[m_] := Sum[Sum[pnpn[n]^j/j, {n,1,m}], {j,1,Infinity}] ppreduced[m_] := Exp[logppreduced[m]] Note that these are increasing functions of m. If we find some that are larger than n2nrecipsum then we can conclude the original product is larger. In[25]:= InputForm[ppapproxtable10 = Table[ppreduced[m], {m,1,10}]] Out[25]//InputForm= {4/3, 18/13, 28125/20306, 7720715625/5574281284, 40051095877840018125/28916500101601742968, 311037947003521544297171294989375/ 224566360270448556138245899597024, 257303112735781697706873542424979340522046371095619375/ 185770334680987315271766907548466081748690434519412224, 5090535356990520743958828490802637081255368479042693424799727037262506694931\ 25/36753168157146013578531065523610752369996787371198452775938443849075962470\ 7072, 96629782385213577755527315475090745315830719668036862913363713717694372\ 3539154430934663228123511502122395625/697657592362105993889141526695312334515\ 430396279207005661133480952594500728389800754298187596407056120338432, 1222241153761054248281966261025369409274584690758326302779868638284392065563\ 8089259422255118350077241514349839708995048550866095718050049489956849375/882\ 44617712116664750138910377647286194885591081164810294385639757572231217489133\ 02294604250917317541588237740724590755567562424488347610728779374592} Furthermore N[ppapproxtable10,50] will reveal that these have stabilized to over 40 decimal places, the first several digits being 1.3850602852... Now we'll have a look at n2nrecipsum. Aside remark: n2nrecipsum is reasonably close to 2-1/Sqrt[2] (I assume this is what was meant above). They agree to between three and four decimal places. It is well known though not elementary (follows from Liouville's theorem, the one on diophantine approximation by rationals, not the one that bounded entire functions are constant) that n2nrecipsum is transcendental. I believe it then follows that it will not be too well approximated by an algebraic of low degree and involving smallish coefficients in its minimal polynomial. I guess I'm being sufficiently vague here that I can assert the veracity, if necessary defining "well approximated" in such a way as to discard any counterexamples that might be proferred. On to the issue at hand. We now observe that it is easy to majorize n2nrecipsum. We can sum to a finite bound m, then replace n^n by n^m thereafter in summing from m+1 to infinity. With this we obtain a sequence of approximations, each element of which is explicitly evaluatable. It is not hard to show that the sequence converges to n2nrecipsum. If we learn that any element of that sequence is less than any ppreduced[j] then we have established that n2nrecipsum<pp. n2nrecipsumenlarged[m_] := Sum[1/n^n, {n,1,m}] + Sum[1/n^m, {n,m+1,Infinity}] InputForm[n2nrecipsumapproxtable10 = Table[n2nrecipsumenlarged[m], {m,2,10}]] Out[39]//InputForm= {5/4 + (-15 + 2*Pi^2)/12, 139/108 + (-251 + 216*Zeta[3])/216, 8923/6912 + (-111845 + 1152*Pi^4)/103680, 27891287/21600000 + (-806108207 + 777600000*Zeta[5])/777600000, 753077249/583200000 + (-332250636263 + 345600000*Pi^6)/326592000000, 620192080073207/480290277600000 + (-774879868932307123 + 768464444160000000*Zeta[7])/768464444160000000, 40644910035811590827/31476303632793600000 + (-248886558707571775009601 + 26230253027328000000*Pi^8)/ 247875891108249600000000, 21600371677519118879091707/16727798278915463577600000 + (-4106541588424891370931874221019 + 4098310578334288576512000000000* Zeta[9])/4098310578334288576512000000000, 67501161497474683459322666743/52274369621610823680000000000 + (-4548726323970653923866830952038699 + 48572569817295272017920000000*Pi^10)/ 4544206769257059173636505600000000} N[n2nrecipsumapproxtable10,20] reveals these have converged to around 8 digits, to 1.291286 or so. As 1.2913<1.385 we have are now quite certain that n2nrecipsum<pp. Along the way we have validated the approximate values for these that were given in the original note. Daniel Lichtblau Wolfram Research

**References**:**another problem with Infinite Products***From:*Roger Bagula <rlbagulatftn@yahoo.com>