Remark: yepp, indeed I misread the (n)-construct in the discussed inner sum not as derivative; sorry. It's sometimes difficult to switch between the consideration of iterates and then of derivatives; sometimes a short remark in the near context, say "// derivatives" may be helpful... <sigh>
The estimate for B_n/n! is about 2*(2pi)^n which is approached early with very good accuracy. Here is the list of ratios B_n/n! * 2*(2pi)^n (only nonzero entries listed) :
This determines the range of convergence for some function, if it is applied to the bernoulli-numbers in the form of the inner sum, depending on their own Taylor/Maclaurin-coefficients and its parameter x.
If - for instance - the function exp(x) is applied, then the Maclaurin-coefficients are all 1. These are cofactored with the bernoulli-numbers divided by factorials: thus the convergence-radius for this function in this application is x<=2*Pi.
If for another function the coefficients grow with some geometric rate, say q, the convergence-radius for that function in this application is x < 2*pi/q
If for another function the Maclaurin-coefficients grow with some hypergeometric rate, then the convergence-radius for that series in this form of application is zero.
Additional remark: for the powerseries of fractional iterates for regular iteration I found in 2007 and 2008[*1,*2], that the growthrate of the maclaurin-coefficients beginning at some finite index k is roughly binomial(n-k,2) (hope I recall right), so asymptotically for n-> inf about n(n-1)/2 =n^2/2-n/2 so their convergence-radius wr to its parameter x is also zero (as originally stated by I.N.Baker , and is also not even Borel-summable, since this grows faster than the factorials)
[update] well, with a second read of this I think one more remark is useful. If we see the above double sum as two-dimensional table (or matrix), where the inner sum runs over a column, then my discussion here concerns only one column (and namely the first one, while the original problem concerns the row-sums and then the column-sum of the rowsums). For the convergence of the whole sum (including the outer sum over all columns) we'll have to extend this consideration about the range of covergence.
[/update].
Gottfried
[*1] see coefficients fractional height 1
[*2] see also coefficients fractional height 2
The estimate for B_n/n! is about 2*(2pi)^n which is approached early with very good accuracy. Here is the list of ratios B_n/n! * 2*(2pi)^n (only nonzero entries listed) :
Code:
0.500000000000
1.64493406685
-1.08232323371
1.01734306198
-1.00407735620
1.00099457513
-1.00024608655
1.00006124814
-1.00001528226
1.00000381729
-1.00000095396
1.00000023845
-1.00000005961
1.00000001490
-1.00000000373
1.00000000093
-1.00000000023
1.00000000006
-1.00000000001
1.00000000000
-1.00000000000
1.00000000000If - for instance - the function exp(x) is applied, then the Maclaurin-coefficients are all 1. These are cofactored with the bernoulli-numbers divided by factorials: thus the convergence-radius for this function in this application is x<=2*Pi.
If for another function the coefficients grow with some geometric rate, say q, the convergence-radius for that function in this application is x < 2*pi/q
If for another function the Maclaurin-coefficients grow with some hypergeometric rate, then the convergence-radius for that series in this form of application is zero.
Additional remark: for the powerseries of fractional iterates for regular iteration I found in 2007 and 2008[*1,*2], that the growthrate of the maclaurin-coefficients beginning at some finite index k is roughly binomial(n-k,2) (hope I recall right), so asymptotically for n-> inf about n(n-1)/2 =n^2/2-n/2 so their convergence-radius wr to its parameter x is also zero (as originally stated by I.N.Baker , and is also not even Borel-summable, since this grows faster than the factorials)
[update] well, with a second read of this I think one more remark is useful. If we see the above double sum as two-dimensional table (or matrix), where the inner sum runs over a column, then my discussion here concerns only one column (and namely the first one, while the original problem concerns the row-sums and then the column-sum of the rowsums). For the convergence of the whole sum (including the outer sum over all columns) we'll have to extend this consideration about the range of covergence.
[/update].
Gottfried
[*1] see coefficients fractional height 1
[*2] see also coefficients fractional height 2
Gottfried Helms, Kassel