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

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.