K. Knopp explains this with the idea of comparision of the transformed series to be summed with the transformation sequence without the series coefficients.
So he says: first, your original series might be notated by \( \sum_{k=0}^\infty a_k \).
With this, you define the sequence of partial sums by \( s_k = \sum_{j=0}^k a_j \) .
Many of the classical summation procedures work on the basic concept of averaging the partial sums \( s_k \) (the simplest ones Hölder and Cesaro summation) .
The Borel-summation can then be understood as the transformation of the \( s_k \) values and the "averaging" of that partial sums by transforming and comparing:
\[ \mathcal B \quad s = \lim_{x \to \infty} { \sum_{k=0}^\infty { x^k \cdot s_k \over k! }\over \sum_{k=0}^\infty { x^k \over k! } } \]
If the values \( a_k \) have factorial growthrate, say \( \mid a_k \mid = k! \) , then as well the \( s_k \) have this growthrate. Now in the numerator-sum we take the transformed \( s_k \) which means they get divided by \( k! \) and thus the series in the numerator has now simply geometric growth. But the partial sums \( s_k \) are not simple factorials, so the resulting transformed series is not exactly geometric, and can thus not be replaced by its known finite value. So we are here at a dead end.
But Knopp explains then, that the Borel-transformation need not only transform by division by \( k! \) but we may introduce a further factor, the "order" of Borel summation, say \( r \).
Then the transformation looks like
\[ \mathcal B_r \quad s = \lim_{x \to \infty} { \sum_{k=0}^\infty { x^{rk} \cdot s_k \over (rk)! }\over \sum_{k=0}^\infty { x^{rk} \over (rk)! } } \]
and with \(r=2\) the sum in the numerator as well as in the denominator becomes convergent for any \( x \), can be evaluated, and then the numerator-sum is "averaged" by the denominator-sum.
Practically using software we deal with truncated series; but the Borel-sum requires that in the terms of the series as well as the coefficient \( x^k \) or \( x^{rk} \) is present, and we know, that for large \( x \) the exponential series increases to huge values before the influence of the denominator begins to diminuish and suppress that \(x^k\) term. So we cannot really compute with arbitrary large \( x \) (which would be required) but only such large values, that our finitely long exponentialseries has dominated that \(x^k \) or \( x^{rk}\) values. So the implementation of this version of the Borel sum is never really perfect
For instance, to sum the halfiterate of the \( \exp(z)-1 \) I needed order \( r=2 \) but could not use \( x \gt 70 \) say, because of having only 256 or 512 transformed terms. Having 1024 terms, one could insert \( x \) with value of around 80 or 100 and can thus achieve more digits precision.
- - - -
P.s.: Practically we need only the proof on concept. After we know that Borel-summation can manage the current growthrate of the \( a_k \) or more precisely that of the \( s_k \) we had of course the possibility not to use \( z=1 \) but the -say- 40th iteration towards 0 by negative height getting \( z_{-40} \lt 1e-10 \), use then for the partial sums \( s_k = \sum_{j=0}^k z_{-40}^k \cdot a_k \) and get nearly exact values for the Borel-sum because \( x \) can now go much farther to \( \infty \) .
I add the chapter 13 of the Knoop book (from where I got this) (in its german original) here as pdf
Knopp UnendlicheReihen Kap13.pdf (Size: 8.68 MB / Downloads: 580)
. The here paraphrased Borel-method is at pg 488.
P.P.s. I have Pari/GP code for testing with that Borel-method, but I think it is too dense to present it at the moment here. I'll see that I can make it better readable and show it then here if needed.
So he says: first, your original series might be notated by \( \sum_{k=0}^\infty a_k \).
With this, you define the sequence of partial sums by \( s_k = \sum_{j=0}^k a_j \) .
Many of the classical summation procedures work on the basic concept of averaging the partial sums \( s_k \) (the simplest ones Hölder and Cesaro summation) .
The Borel-summation can then be understood as the transformation of the \( s_k \) values and the "averaging" of that partial sums by transforming and comparing:
\[ \mathcal B \quad s = \lim_{x \to \infty} { \sum_{k=0}^\infty { x^k \cdot s_k \over k! }\over \sum_{k=0}^\infty { x^k \over k! } } \]
If the values \( a_k \) have factorial growthrate, say \( \mid a_k \mid = k! \) , then as well the \( s_k \) have this growthrate. Now in the numerator-sum we take the transformed \( s_k \) which means they get divided by \( k! \) and thus the series in the numerator has now simply geometric growth. But the partial sums \( s_k \) are not simple factorials, so the resulting transformed series is not exactly geometric, and can thus not be replaced by its known finite value. So we are here at a dead end.
But Knopp explains then, that the Borel-transformation need not only transform by division by \( k! \) but we may introduce a further factor, the "order" of Borel summation, say \( r \).
Then the transformation looks like
\[ \mathcal B_r \quad s = \lim_{x \to \infty} { \sum_{k=0}^\infty { x^{rk} \cdot s_k \over (rk)! }\over \sum_{k=0}^\infty { x^{rk} \over (rk)! } } \]
and with \(r=2\) the sum in the numerator as well as in the denominator becomes convergent for any \( x \), can be evaluated, and then the numerator-sum is "averaged" by the denominator-sum.
Practically using software we deal with truncated series; but the Borel-sum requires that in the terms of the series as well as the coefficient \( x^k \) or \( x^{rk} \) is present, and we know, that for large \( x \) the exponential series increases to huge values before the influence of the denominator begins to diminuish and suppress that \(x^k\) term. So we cannot really compute with arbitrary large \( x \) (which would be required) but only such large values, that our finitely long exponentialseries has dominated that \(x^k \) or \( x^{rk}\) values. So the implementation of this version of the Borel sum is never really perfect
For instance, to sum the halfiterate of the \( \exp(z)-1 \) I needed order \( r=2 \) but could not use \( x \gt 70 \) say, because of having only 256 or 512 transformed terms. Having 1024 terms, one could insert \( x \) with value of around 80 or 100 and can thus achieve more digits precision.
- - - -
P.s.: Practically we need only the proof on concept. After we know that Borel-summation can manage the current growthrate of the \( a_k \) or more precisely that of the \( s_k \) we had of course the possibility not to use \( z=1 \) but the -say- 40th iteration towards 0 by negative height getting \( z_{-40} \lt 1e-10 \), use then for the partial sums \( s_k = \sum_{j=0}^k z_{-40}^k \cdot a_k \) and get nearly exact values for the Borel-sum because \( x \) can now go much farther to \( \infty \) .
I add the chapter 13 of the Knoop book (from where I got this) (in its german original) here as pdf
Knopp UnendlicheReihen Kap13.pdf (Size: 8.68 MB / Downloads: 580)
. The here paraphrased Borel-method is at pg 488.P.P.s. I have Pari/GP code for testing with that Borel-method, but I think it is too dense to present it at the moment here. I'll see that I can make it better readable and show it then here if needed.
Gottfried Helms, Kassel