Ways to make a series diverge "faster" to show divergence
Here is a dataexample using the Euler-summation of negative instead of positive orders. I used the slowly divergent series $1+1/2+1/3+1/4+...$ and the sequences of partial sums using Eulersummation $ES(0)$ (direkt summation= no transformation) , Eulersummation $ES(-0.5)$ which has negative order and should accelerate divergence and Eulersummation $ES(-0.9)$ which accelerates divergence but even "overtunes" it: it makes it look like an alternating series. (All computations based on 32 elements) :
ES(0)(direct) ES(-0.5) ES(-0.9)
1.00000000000 1.00000000000 1.00000000000
1.50000000000 2.00000000000 6.00000000000
1.83333333333 2.33333333333 -5.66666666667
2.08333333333 2.66666666667 49.3333333333
2.28333333333 2.86666666667 -245.666666667
2.45000000000 3.06666666667 1526.00000000
2.59285714286 3.20952380952 -9861.85714286
2.71785714286 3.35238095238 67007.4285714
2.82896825397 3.46349206349 -471076.460317
2.92896825397 3.57460317460 3403128.53968
3.01987734488 3.66551226551 -25125107.3694
3.10321067821 3.75642135642 188836662.782
3.18013375513 3.83334443334 -1440564509.14
3.25156232656 3.91026751027 11129101675.0
3.31822899323 3.97693417693 -86914294560.5
3.38072899323 4.04360084360 685177450795.
3.43955252264 4.10242437301 -5.44613935055E12
3.49510807820 4.16124790242 4.36043950602E13
3.54773965714 4.21387948137 -3.51381487300E14
3.59773965714 4.26651106032 2.84800415982E15
3.64535870476 4.31413010794 -2.32041361096E16
3.69081325022 4.36174915556 1.89949738822E17
3.73429151109 4.40522741643 -1.56161905953E18
3.77595817775 4.44870567730 1.28888235269E19
3.81595817775 4.48870567730 -1.06760841088E20
3.85441971622 4.52870567730 8.87251757254E20
3.89145675325 4.56574271433 -7.39618656227E21
3.92717103897 4.60277975137 6.18296908223E22
3.96165379759 4.63726250999 -5.18235419676E23
3.99498713092 4.67174526861 4.35431150851E24
4.02724519544 4.70400333313 -3.66693900482E25
4.05849519544 4.73626139764 3.09468091836E26
However, I do not really think that this can be a general useful tool: since also convergent series might look like divergent by such "inverse" transformations. Here I used $1+1/2^2+1/3^2+1/4^2+...$
ES(0)(direct) ES(-0.5) ES(-0.9)
1.00000000000 1.00000000000 1.00000000000
1.25000000000 1.50000000000 3.50000000000
1.36111111111 1.44444444444 -7.88888888889
1.42361111111 1.55555555556 57.1111111111
1.46361111111 1.52888888889 -352.888888889
1.49138888889 1.58000000000 2402.38888889
1.51179705215 1.56390022676 -16936.9478458
1.52742205215 1.59383219955 123212.235828
1.53976773117 1.58289745528 -917619.148652
1.54976773117 1.60275636180 6963787.31960
1.55803219398 1.59477262837 -53665499.9384
1.56497663842 1.60900149714 418884302.009
1.57089379818 1.60287884486 -3305102741.43
1.57599583900 1.61362133802 26320630606.8
1.58044028344 1.60875562173 -211296315925.
1.58434653344 1.61717979015 1.70819287210E12
1.58780674106 1.61320690798 -1.38954751976E13
1.59089316081 1.62000633266 1.13658899049E14
1.59366324391 1.61669272000 -9.34278213695E14
1.59616324391 1.62230655034 7.71398168189E15
1.59843081761 1.61949494420 -6.39481412903E16
1.60049693331 1.62421545194 5.32067131888E17
1.60238729248 1.62179578230 -4.44177875633E18
1.60412340359 1.62582540701 3.71945515959E19
1.60572340359 1.62371815228 -3.12340250305E20
1.60720269353 1.62720177203 2.62971858850E21
1.60857443564 1.62534794030 -2.21942324134E22
1.60984994585 1.62839210680 1.87735425152E23
1.61103900649 1.62674694346 -1.59133257136E24
1.61215011760 1.62943185453 1.35152568303E25
1.61319070033 1.62796074625 -1.14995824956E26
1.61416726283 1.63034797478 9.80133223927E26
With $ES(-0.5)$ we get a sequence which lingers around at least in the near of the known result - but who would prognose that this extrapolates actually to a certain limit. And the sequence of partial sums of the $ES(-0.9)$-acceleration looks undoubtedly divergent...
So I think with the standard tools for acceleration/divergent summation taken only in an obviously/naively inverted way we are not yet really succeeding/proceeding in the desired direction...