Trying for several digits of DanA's Out[197]= Sum[Product[1 - 1/Prime[k], {k, 1, -1 + n}]/Prime[n], {n, 1, Infinity}] (No need to special-case n=1 -- Mma does the right thing on empty products. And the *wrong* thing when they're of negative length!) In[198]:= shanks[L_List] := (#1*#3 - #2^2)/(#1 - 2*#2 + #3) &[Drop[L, -2], L[[2 ;; -2]], Drop[L, 2]] Take exponentially longer segments of the series: In[187]:= numMProd[ Table[MProd[ MatrixForm[{{1 - 1./Prime[i], 1/Prime[i]}, {0, 1}}], {i, 2^n + 1, 2^(n + 1)}], {n, 0, 11}]] Out[187]= {{{0.666667, 1/3}, {0, 1}}, {{0.685714, 0.314286}, {0., 1.}}, {{0.74823, 0.25177}, {0., 1.}}, {{0.795719, 0.204281}, {0., 1.}}, {{0.83033, 0.16967}, {0., 1.}}, {{0.856913, 0.143087}, {0., 1.}}, {{0.876387, 0.123613}, {0., 1.}}, {{0.891033, 0.108967}, {0., 1.}}, {{0.902836, 0.0971637}, {0., 1.}}, {{0.912323, 0.0876769}, {0., 1.}}, {{0.920145, 0.079855}, {0., 1.}}, {{0.926663, 0.0733374}, {0., 1.}}} Restore the first term, and accumulate: In[188]:= FoldList[Dot, {{1, 1}/2, {0, 1}}, %] Out[188]= {{{1/2, 1/2}, {0, 1}}, {{0.333333, 0.666667}, {0., 1.}}, {{0.228571, 0.771429}, {0., 1.}}, {{0.171024, 0.828976}, {0., 1.}}, {{0.136087, 0.863913}, {0., 1.}}, {{0.112997, 0.887003}, {0., 1.}}, {{0.0968287, 0.903171}, {0., 1.}}, {{0.0848595, 0.915141}, {0., 1.}}, {{0.0756126, 0.924387}, {0., 1.}}, {{0.0682658, 0.931734}, {0., 1.}}, {{0.0622804, 0.93772}, {0., 1.}}, {{0.057307, 0.942693}, {0., 1.}}, {{0.0531043, 0.946896}, {0., 1.}}} The partial sums are in the upper right [[1,2]]: In[189]:= shanks[#[[1, 2]] & /@ %] Out[189]= {0.948718, 0.899118, 0.917897, 0.932005, 0.94094, 0.949258, \ 0.955796, 0.96014, 0.964033, 0.967136, 0.969815} Second tier: In[190]:= shanks[%] Out[190]= {0.912739, 0.974621, 0.956372, 1.06137, 0.97982, 0.968741, \ 0.997558, 0.979348, 0.986716} Third (losing both ends each time.) In[191]:= shanks[%] Out[191]= {0.960528, 0.971919, 1.01547, 0.966999, 0.976743, 0.986399, \ 0.984594} Fourth. In[192]:= shanks[%] Out[192]= {0.956494, 0.99253, 0.975112, 2.0492, 0.984878} A tall pole in the tent! The immediate explanation is that 0.966999, 0.976743, 0.986399 is nearly in arithmetic progression, provoking division by nearly 0. In fact, the irregularity of the primes makes the process so noisy that the second differences change sign wantonly, and this whole computation is meaningless. But suppose we smooth it out, replacing Prime[n] with n*Log[n]. That's an interesting sum, too. Suppose we sum 10^6 terms directly, and Shanks the tail: numMProd[Table[ MProd[MatrixForm[{{1 - 1`33/(n*Log[n]), 1/(n*Log[n])}, {0, 1}}], {n, 2^k + 1000000, 999999 + 2^(k + 1)}], {k, 0, 11}]]; (Here, numMProd just performs the matrix products, to 33 places due to the 1`33.) Form the sequence of exponentially longer partial sums: FoldList[Dot, IdentityMatrix[2], %]; Run one Shanks: shanks[#[[1, 2]] & /@ %] {-7.23825846114180023820963784936630936956*10^-8, \ -7.2383081775534954986936726671956993755*10^-8, \ -7.238507042592270807744603740319705184*10^-8, \ -7.239302497883386891719990656592430349*10^-8, \ -7.24248428013694135359940241455532980*10^-8, \ -7.25521109787942075472466273513777568*10^-8, \ -7.3061158789240489622769357685310492*10^-8, \ -7.509715087677159821674775767630926*10^-8, \ -8.323952662885734390860079101633870*10^-8, \ -1.157962990187933870110393101592588*10^-7, \ -2.459217059666940024454702903981525*10^-7} How can these conceivably be <0?? Try the 2nd order Shanks: shanks2[#[[1, 2]] & /@ %198] {-7.2382418887852421433705793226704*10^-8, \ -7.238241887452633043864365057607*10^-8, \ -7.238241876791840693390633496226*10^-8, \ -7.238241791506788978747515379514*10^-8, \ -7.23824110924696740587213638954*10^-8, \ -7.23823565149782811295872404833*10^-8, \ -7.23819199477433435746658414992*10^-8, \ -7.2378428252583618557144316956*10^-8, \ -7.2350508161419247635566250253*10^-8} Still negative! The only glimmer of sanity comes from summing bursts of length 3^n, shanks[#[[1, 2]] & /@ %] {-7.23827917631153038724969102081745002169*10^-8, \ -7.2385774734354090466183845664296486922*10^-8, \ -7.241262106902162871015196793342498037*10^-8, \ -7.26542271070426051438560418211001551*10^-8, \ -7.4828385233749802945842107930570293*10^-8, \ -9.438781720404053555835539434832801*10^-8, \ -2.702074800500521053964558787965094*10^-7, \ -1.846816420328610065496655139531113*10^-6, \ -0.00001588394699143661453072982592933523, \ -0.0001383570856200631342075355905173333, \ -0.001152755525647733026707277401399129} and the using Shanks2: shanks2[#[[1, 2]] & /@ %%] {-7.23824188680716125090267195281569*10^-8, \ -7.2382418304287692662035666190882*10^-8, \ -7.238240308325929783457067074808*10^-8, \ -7.238199220757133558198054009671*10^-8, \ -7.237090600928513738314351741322*10^-8, \ -7.20721785747766461202604502914*10^-8, \ -6.40543875690906529608871726514*10^-8, 1.48716801605978891479361949759*10^-7, 5.62589299046691128953938459235*10^-6} I have never before seen this wrong sign behavior on a smooth sequence. Unnerving. --rwg Does anyone have a nice little Richardson accelerator to share? Is it time to give up and beg Dan for his answer?