# Euler Transform elementary Proof

In this webpage Computing the Digits in π there is a proof of the Euler Transform (page 22).

The proof there relies on measure theory and Lebesgue integration, I haven't studied that yet.

In page 22 there is the following statement:

Euler didn’t actually prove any general theorems about this transformation. He did use it in several specific cases, where he could show that it really did converge to the original sum, and converged much more quickly.

I was wondering if anyone knows any elementary proof of this transformation or a proof for a particular series ?

I don't find many information of this transformation online, a resource recommendation is welcome

• There is a scan of K.Knopp's somehow classical book on infinite series, with chapt. 13 on divergent series and Euler-summation online available, however I only know the access to the german version (@ digicenter Goettingen) and don't know if the english translation is also accessible. Perhaps via google books; but just talking about this I know that G.H.Hardy's book is (partially) readable via that place. – Gottfried Helms Nov 20 '15 at 18:28
• What's the precise name of the book @GottfriedHelms I found this in wiki :en.m.wikipedia.org/wiki/Konrad_Knopp#Books – Keith Nov 20 '15 at 20:35
• I know only that of 1928 with the "theory of..." in the title (and is the 4'th edition of 1947). Perhaps the 1956 book is even a conceptual update but I've not yet looked inside it. – Gottfried Helms Nov 20 '15 at 20:44
• Thanks for the quite nice reference anyway. – Han de Bruijn Nov 21 '15 at 12:55
• In his 1930 article Helmut Hasse develops a variant for divergent summation of the zeta-series, which initial part uses an element of the Euler-summation; he formulates his method with a proof. Why this might be relevant for you is that K.Knopp had entered the scene and wrote that he had the last step already shown with means of the full Euler-summation, and H. Hasse acknowledges this in a footnote. So this might be usable as another proof when composed. The Hasse article is unfortunately again in german, accessible via digicenter Göttingen ("Ein summierungsverfahren für die Zeta-Funktion") – Gottfried Helms Nov 25 '15 at 21:47

This is not an answer (I don't know the formal proof) but a comment because the power of the Euler-summation for series like this is much impressive but often not really known.

Here is a table of the progression to the final value without and with Euler-summation. Euler-summation can have "orders", which intuitively means, iterates (but can be interpolated to fractional orders). Here is the table using Euler-summation "to order (0.5)" :

the individual         partial          distance to     partial sums       distance to pi/4
terms of the series    sums             pi/4            by Euler-summ.
-----------------------------------------------------------------------------------------------
1.00000000000   1.00000000000     0.214601836603   1.00000000000            0.214601836603
-0.333333333333  0.666666666667    -0.118731496731  0.777777777778         -0.00762038561967
0.200000000000  0.866666666667    0.0812685032692  0.792592592593          0.00719442919514
-0.142857142857  0.723809523810   -0.0615886395879  0.784832451499        -0.000565711898330
0.111111111111  0.834920634921    0.0495224715232  0.785851459926         0.000453296528086
-0.0909090909091  0.744011544012   -0.0413866193859  0.785350269301       -0.0000478940965617
0.0769230769231  0.820934620935    0.0355364575372  0.785432796132        0.0000346327349363
-0.0666666666667  0.754267954268   -0.0311302091295  0.785393836971      -0.00000432642610791
0.0588235294118  0.813091483680    0.0276933202823  0.785401073569       0.00000291017167712
-0.0526315789474  0.760459904732   -0.0249382586651  0.785397756972     -0.000000406425094661
0.0476190476190  0.808078952351    0.0226807889539  0.785398422239      0.000000258841333253
-0.0434782608696  0.764600691482   -0.0207974719156  0.785398124198    -0.0000000391999284656
0.0400000000000  0.804600691482    0.0192025280844  0.785398187306     0.0000000239087486163
-0.0370370370370  0.767563654445   -0.0178345089527  0.785398159544   -0.00000000385353035100
0.0344827586207  0.802046413065    0.0166482496680  0.785398165666    0.00000000226867304697
-0.0322580645161  0.769788348549   -0.0156098148481  0.785398163013  -0.000000000384322878997
0.0303030303030  0.800091378852    0.0146932154549  0.785398163617         2.19652484372E-10
-0.0285714285714  0.771519950281   -0.0138782131165  0.785398163359        -3.87659406085E-11
0.0270270270270  0.798546977308    0.0131488139105  0.785398163419         2.16018278683E-11
-0.0256410256410  0.772905951667   -0.0124922117305  0.785398163394        -3.94612015099E-12
0.0243902439024  0.797296195569    0.0118980321720  0.785398163400         2.15112513145E-12
-0.0232558139535  0.774040381616   -0.0113577817815  0.785398163397        -4.04724379093E-13
0.0222222222222  0.796262603838    0.0108644404407  0.785398163398         2.16404904436E-13
-0.0212765957447  0.774986008093   -0.0104121553040  0.785398163397        -4.17725596145E-14
0.0204081632653  0.795394171359   0.00999600796131  0.785398163397         2.19558323752E-14
-0.0196078431373  0.775786328222  -0.00961183517595  0.785398163397        -4.33468728127E-15
0.0188679245283  0.794654252750   0.00925608935236  0.785398163397         2.24358184761E-15
-0.0181818181818  0.776472434568  -0.00892572882946  0.785398163397        -4.51894496246E-16
0.0175438596491  0.794016294217   0.00861813081966  0.785398163397         2.30671631889E-16
-0.0169491525424  0.777067141675  -0.00833102172271  0.785398163397        -4.73009122777E-17
0.0163934426230  0.793460584298   0.00806242090024  0.785398163397         2.38422950230E-17
-0.0158730158730  0.777587568425  -0.00781059497278  0.785398163397        -4.96870530081E-18


Note that this "order of 0.5" seems to be optimal; the simple Euler-summation (which were "of order 1") accelerates not so spectacular.