03/15/2008, 09:13 AM
Occasionally I reread older threads, sometimes one finds ome gems in them of which one got not aware earlier. It seems to me, it might be useful to give this thread some conclusive response to also help other occasional readers to sort out and weight the scattered examples, arguments and conclusions.
In the table below I document the first 64 terms of the powerseries for f°(1/2)(x) where f(x)=exp(x)-1. The powerseries for f°(1/2)(x) was destilled using the matrix-logarithm of the matrix S2 (rescaled Stirling numbers 2'nd kind), which provides the powerseries for f(x)=exp(x)-1 .
The first column contains the terms in rational representation and shows the same coefficients, which Henryk provided as an example. However, Henryk's example ended at some small index, but the continuation shows, that the terms of the powerseries actually diverge from a certain index. The rate of growth of terms is then roughly hypergeometric (guessed by inspection).
The second column shows the same terms in real arithmetic.
The third column shows the partial sums up to <rownumber> of terms; this also implies, that simply x=1 is assumed here. After a good approximation this sequence diverges, too, so indeed f°(1/2)(1) cannot simply be summed to a limit value.
The last column shows partial sums using an extension of Euler-summation. Since common Euler-summation of any order can only sum series with geometric growth, I used a variant, which (hopefully) compensates this hypergeometric growth (method not documented/discussed yet) and indeed seems to be able to transform the series into a convergent one, providing the result of
f°(1/2)(1)~ 1.2710274
Gottfried
In the table below I document the first 64 terms of the powerseries for f°(1/2)(x) where f(x)=exp(x)-1. The powerseries for f°(1/2)(x) was destilled using the matrix-logarithm of the matrix S2 (rescaled Stirling numbers 2'nd kind), which provides the powerseries for f(x)=exp(x)-1 .
The first column contains the terms in rational representation and shows the same coefficients, which Henryk provided as an example. However, Henryk's example ended at some small index, but the continuation shows, that the terms of the powerseries actually diverge from a certain index. The rate of growth of terms is then roughly hypergeometric (guessed by inspection).
The second column shows the same terms in real arithmetic.
The third column shows the partial sums up to <rownumber> of terms; this also implies, that simply x=1 is assumed here. After a good approximation this sequence diverges, too, so indeed f°(1/2)(1) cannot simply be summed to a limit value.
The last column shows partial sums using an extension of Euler-summation. Since common Euler-summation of any order can only sum series with geometric growth, I used a variant, which (hopefully) compensates this hypergeometric growth (method not documented/discussed yet) and indeed seems to be able to transform the series into a convergent one, providing the result of
f°(1/2)(1)~ 1.2710274
Gottfried
Code:
´
0 0 0 0
1 1 1 0.45454545
1/4 0.25000000 1.2500000 0.74050542
1/48 0.020833333 1.2708333 0.92356508
0 0.E-810 1.2708333 1.0421115
1/3840 0.00026041667 1.2710938 1.1195105
-7/92160 -0.000075954861 1.2710178 1.1703562
1/645120 0.0000015500992 1.2710193 1.2039209
53/3440640 0.000015404111 1.2710347 1.2261669
-0.0000090745391 -0.0000090745391 1.2710257 1.2409615
-0.000000082819971 -0.000000082819971 1.2710256 1.2508300
0.0000036074073 0.0000036074073 1.2710292 1.2574303
-0.0000016951497 -0.0000016951497 1.2710275 1.2618553
-0.0000013308992 -0.0000013308992 1.2710262 1.2648287
0.0000017752144 0.0000017752144 1.2710279 1.2668308
0.00000037035398 0.00000037035398 1.2710283 1.2681815
-0.0000019147568 -0.0000019147568 1.2710264 1.2690944
0.00000034467343 0.00000034467343 1.2710267 1.2697124
0.0000024191341 0.0000024191341 1.2710292 1.2701316
-0.0000014770587 -0.0000014770587 1.2710277 1.2704162
-0.0000036046260 -0.0000036046260 1.2710241 1.2706099
0.0000042603060 0.0000042603060 1.2710283 1.2707418
0.0000061940178 0.0000061940178 1.2710345 1.2708318
-0.000012625293 -0.000012625293 1.2710219 1.2708933
-0.000011736089 -0.000011736089 1.2710102 1.2709353
0.000041395229 0.000041395229 1.2710516 1.2709641
0.000022203030 0.000022203030 1.2710738 1.2709839
-0.00015310857 -0.00015310857 1.2709207 1.2709974
-0.000027832787 -0.000027832787 1.2708928 1.2710067
0.00064101866 0.00064101866 1.2715339 1.2710131
-0.00011130752 -0.00011130752 1.2714225 1.2710175
-0.0030302667 -0.0030302667 1.2683923 1.2710206
0.0016766696 0.0016766696 1.2700690 1.2710227
0.016095115 0.016095115 1.2861641 1.2710241
-0.015708416 -0.015708416 1.2704556 1.2710251
-0.095480465 -0.095480465 1.1749752 1.2710258
0.13948961 0.13948961 1.3144648 1.2710263
0.62852065 0.62852065 1.9429854 1.2710267
-1.2769417 -1.2769417 0.66604378 1.2710269
-4.5595640 -4.5595640 -3.8935202 1.2710270
12.402772 12.402772 8.5092522 1.2710272
36.185455 36.185455 44.694707 1.2710272
-129.30559 -129.30559 -84.610888 1.2710273
-311.60844 -311.60844 -396.21933 1.2710273
1453.7164 1453.7164 1057.4971 1.2710274
2883.7550 2883.7550 3941.2521 1.2710274
-17648.606 -17648.606 -13707.354 1.2710274
-28323.267 -28323.267 -42030.620 1.2710274
231312.84 231312.84 189282.22 1.2710274
289837.71 289837.71 479119.93 1.2710274
-3269336.0 -3269336.0 -2790216.0 1.2710274
-2992168.6 -2992168.6 -5782384.6 1.2710274
49750634. 49750634. 43968250. 1.2710274
28980063. 28980063. 72948313. 1.2710274
-8.1361647E8 -8.1361647E8 -7.4066816E8 1.2710274
-2.0196159E8 -2.0196159E8 -9.4262976E8 1.2710274
1.4271686E10 1.4271686E10 1.3329057E10 1.2710274
-1.3254909E9 -1.3254909E9 1.2003566E10 1.2710274
-2.6797851E11 -2.6797851E11 -2.5597494E11 1.2710274
1.1931979E11 1.1931979E11 -1.3665515E11 1.2710274
5.3756367E12 5.3756367E12 5.2389815E12 1.2710274
-4.3701305E12 -4.3701305E12 8.6885108E11 1.2710274
-1.1497780E14 -1.1497780E14 -1.1410895E14 1.2710274
1.3795199E14 1.3795199E14 2.3843037E13 1.2710274
Gottfried Helms, Kassel