su = 0! - 1! + 2! - 3! + 4! - ... + ...

L Euler found a meaningful interpretation using integrals assigning
it a value of about 0.596347…
Also the Borel-summation assigns the same value to this.

Studying the triangle of Eulerian numbers I came across the
idea to use this matrix for a summation, decomposing the entries
of the matrix into geometric series and derivatives.
Not much sophisticated reasoning about range of convergence
included, but it finds the correct value.

See:
  http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf

Chap. 2.2 and 2.3

How could this made waterproof?

TIA -

        fsum(-1) =        0.596347362323
        fsum(-2) =        0.461455316242
        fsum(-3) =        0.385602012137
        fsum(-4) =        0.335221361210
        fsum(-5) =        0.298669749329
        fsum(-6) =        0.270633013639
        fsum(-7) =        0.248281352547
        fsum(-8) =        0.229947781627
        fsum(-9) =        0.214577094581
        fsum(-10) =       0.201464233646
        fsum(-11) =       0.190117766778
        fsum(-12) =       0.180183310425
        …     …

Could someone check this using the integral-formula?

Hmm...

-exp(1)*Ei(-1) = .5963473622
-(1/2)*exp(1/2)*Ei(-1/2) = .4614553164
-(1/3)*exp(1/3)*Ei(-1/3) = .3856020120
-(1/4)*exp(1/4)*Ei(-1/4) = .3352213612
-(1/5)*exp(1/5)*Ei(-1/5) = .2986697494
-(1/6)*exp(1/6)*Ei(-1/6) = .2706330136
-(1/7)*exp(1/7)*Ei(-1/7) = .2482813514
-(1/8)*exp(1/8)*Ei(-1/8) = .2299477818
-(1/9)*exp(1/9)*Ei(-1/9) = .2145771028
-(1/10)*exp(1/10)*Ei(-1/10) = .2014642544
-(1/11)*exp(1/11)*Ei(-1/11) = .1901177930
-(1/12)*exp(1/12)*Ei(-1/12) = .1801833179

Also I see that the coefficients in my example (the colsums of
the Euler-triangle) are not well suited for Cesaro/Euler-summation,
and I assume that the differences from the 8'th digit on are due
to weak performance of the "sumalt"-procedure in Pari/GP for this
problem.
The results with my implementation of Eulersummation agree with the
sumalt-values, and because I have control over the partial sums
I can also explicitely see the difficulties with the convergence
of that partial sums using that summation: convergence is simply
poorly accelerated. So I conclude this is the source of the problem
in Pari/GP' "sumalt".

Well, meanwhile I have some examples, where such a decomposition
into divergent geometric series and reordering summation works
for the divergent case, and also another example, where it does
not. What's the critical point? I've read G.H.Hardy and K.Knopp
few years ago and may not have realized the relevance of some
related chapters there.

Could someone give some more hint?

Gottfried Helms

----------------

where it worked: (for instance) summing of Bell-numbers/
alternating sum of columns of Stirling-matrices,
see http://go.helms-net.de/math/binomial/04_5_SummingBellStirling.pdf

   V(x) represent a columnvector of consecutive powers of x
        (a "Vandermondevector")
   F    the vector of factorials [0!,1!,2!,...], dF when used
        as diagonalmatrix,
   E    the Eulermatrix in lower triangular form,
   ~    the symbol for transposition

then first, we have according to the introductional example

       E*V(1) = F

the factorials as results of rowsums.

If we premultiply that with the inverse factorial, then this gives
the unit-vector:

       dF^-1 * F = V(1)

and the unit-vector premultiplied by a vandermonde-row-vector
with the quotient q of a geometric series evaluates to just that
geometric series in closed form; let's use q=1/2 first:

   V(1/2)~ * V(1) =  2

If we use q=-1 then this is a divergent expression

   V(-1)~ * V(1)  = 1/2   // Cesaro/Euler-summation

But if we dissolve the V(1)-vector we get - formally:

   V(-1)~ * ( dF^-1 * F ) = ???
   V(-1)~ * ( dF^-1 * (E * V(1)) ) = ???

and change order of summation

  ( V(-1)~ * dF^-1 * E ) * V(1) = ???
  (    AS(-1) ~ )        * V(1) = ???

Now let's look at the lhs; the inverse factorials
premultiplied to the Eulerian triangle gives strongly
decreasing values in the intermediate result-triangle and
the premultiplication by the V(-1)-vector has nearly the
same rate of convergence as the exponential series - at
least in the first few columns, obviously.

The first few coefficients of AS(-1) are then

  [0.36787944, 0.13533528, 0.0011826310, -0.0047367048,
   0.00015701391, 0.00020692553, -0.000017334505, -0.0000087610541,
   0.0000012416906, 0.00000034713099, -0.000000075195503,
  -0.000000012470560,...]

and postmultiplied with the V(1)-vector we get the partial sums for
up to 13 terms:

  0.36787944
  0.50321472
  0.50439736
  0.49966065
  0.49981766
  0.50002459
  0.50000726
  0.49999849
  0.49999974
  0.50000008
  0.50000001
  0.50000000
  0.50000000
  ...
For positive q we get ahead with q=0.75 and arrive at 4.000000 with 8 decimals
in the 30'th partial sum;

  (V(0,75) *  dF^-1 * E ) * V(1) -> 4.000

and surely for q=1 we get unresolvable divergence. The first
few terms of AS(1) are (using "sumalt" in Pari/GP)

 [2.7182818, 1.9524924, 1.9957914, 2.0000389, 2.0000576,
  2.0000051, 1.9999996, 1.9999999, 2.0000000, 2.0000000,
  2.0000000, 2.0000000]

which very likely continues for the following terms and the
sum of all terms in AS(1) diverges then to infinity.

But for negative q we can do well: for q=-2

  (V(-2) *  dF^-1 * E ) * V(1)

we get the first few terms in AS(-2)

  [0.13533528, 0.15365092, 0.057425669, -0.0042317431, -0.0092431540,
  -0.0010507275, 0.0012603255, 0.00038236770, -0.00013302980, -0.000082748221,
  0.0000069120015, 0.000014186455]

and the first few partial sums are

  0.13533528
  0.28898621
  0.34641187
  0.34218013
  0.33293698
  0.33188625
  0.33314658
  0.33352894
  0.33339591
  0.33331316
  0.33332008
  0.33333426
  0.33333558
  0.33333357
  0.33333302
  0.33333324
  0.33333337
  0.33333335
  0.33333333

-----------------------------------------------------------

Now it would be good to have the exact range of convergence
for the column-sums. The first two columns are easy: for a
left-multiplication with a vandermonde-vector they provide
infinite range of convergence because of the reciprocal factorials.
But even if the range of convergence for a single column
would be infinite, then the (row-) sum of the column-sums need not
be convergent. This seem to happen at least for q>=1

So I guess with that rough sketch, that we have convergence/summability
for the whole range -inf< q <1 and this agrees also with the ability
to sum the alternating factorial series

Nice exercise/example for the divergent summation stuff - isn't it?

Could this be put to more precision?

 sum k=0..inf  ( AS(1)[k] - 2 ) ->  2/3

If I do alternating summation (with Euler-sum) I get

 sum k=0..inf  ( (-1)^k*AS(1)[k] ) ->  1.761594156  // Euler-summation

which looks like

0.7615941559557648 = (h042) tanh(1)
(by Plouffe's inverter)

  AS(1)*V(-1) = 1 + tanh(1)   // Euler-summation

Strange...