Intereting Posts

Proving $\frac{d}{dx}x^2=2x$ by definition
General and Simple Math Problem.
Proving by strong induction that $\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)…(n+d-1) $
Compendium(s) of Elementary Mathematical Truths
Cardinality of equivalence relations in $\mathbb{N}$
The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
What is the difference between writing f and f(x)?
Given point $P=(x,y)$ and a line $l$, what is a general formula for the reflection of $P$ in $l$
Factorize polynomial over $GF(3)$
Can someone explain curvature in simple terms
How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms?
Name Drawing Puzzle
Extension of a group homomorphism
A series expansion for $\cot (\pi z)$
a set of functions that are pointwise equicontinuous but not uniformly equicontinuous, supposing the domain of f is noncompact

I’ve discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) without conclusive answers)); now I would like to formalize/make waterproof one heuristic.

Consider some (divergent) series $S = K + a + b + c + d + … $. To assign a finite sum to it I’m trying to use the matrix of the Eulerian numbers (let’s call it Eulermatrix for now) to do something in the spirit of Abel- or possibly(?) Borel-summation:

I define the function $$f(x) = K + ax + bx^2 + cx^3 + dx^4 + … $$ with the goal to arrive at a valid/meaningful result for the case that $x \to 1$. The use of the Eulermatrix (see end of the post) implies then the introduction of a transform of $f(x)$ in the form $$ g(x) = K + ax + bx^2/2! + cx^3/3! + dx^4/4! + … $$

If I now evaluate the sequence of partial sums according to the decomposition of the Eulerian numbers in combination with the reciprocal factorials I arrive at the form

$$ s_n(x) = \sum_{k=0}^{n-1} (-1)^k ((n-k)x) ^k {g^{(k)}((n-k)x) \over k!} \qquad n=1 \ldots \infty \tag1$$

and if this converges to a finite value then I assume this as the (divergent) sum of the series and assume $$S = \lim _{n \to \infty} s_n(1)$$

I get meaningful results for series which have growth rates like geometric series with $q \lt 1$ and I think that the formalism of conversion into a double-sum and changing order of evaluation using the Eulermatrix gives also the range for $q$ as $- \infty \lt q \lt 1$ (because the $g(x)$ function is then entire). I can even evaluate the classical series $0! – 1! + 2! – 3! + \ldots – $ to which already L. Euler assigned the value of about $0.596…$ which is now called Gompertz constant.

So I’m confident, that my general reasoning using the Eulermatrix is valid.

- Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
- Please verify my proof of: There is no integer $\geq2$ sum of squares of whose digits equal the integer itself.
- If $n$ is squarefree, $k\ge 2$, then $\exists f\in\Bbb Z : f(\overline x)\equiv 0\pmod n\iff \overline x\equiv \overline 0\pmod n$
- Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction.
- $fg\in L^1$ for every $g\in L^1$ prove $f\in L^{\infty}$
- Proof that a certain entire function is a polynomial

But now: what I’m **asking here** is about the expression for $s_n(x)$ which reminds me of the Maclaurin-expression (which expresses a function by its derivatives at zero), but which I’ve not seen yet. Is that a valid transform? And up to which growth rate can this method handle series (I think, preferably alternating series)?

The Eulermatrix with a rowscaling by reciprocal factorials:

$$ \small \begin{bmatrix}

1 & . & . & . & . & . \\

{1 \over 1!} & . & . & . & . & . \\

{1 \over 2!} & {1 \over 2!} & . & . & . & . \\

{1 \over 3!} & {4 \over 3!} & {1 \over 3!} & . & . & . \\

{1 \over 4!} & {11 \over 4!} & {11 \over 4!} & {1 \over 4!} & . & . \\

{1 \over 5!} & {26 \over 5!} & {66 \over 5!} & {26 \over 5!} & {1 \over 5!} & .\\

\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots

\end{bmatrix} $$

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- Is an isometry necessarily surjective?
- Find the Roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$
- How to prove $3^\pi>\pi^3$ using algebra or geometry?
- Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$
- How is the Riemann zeta function zero at the negative even integers?
- Visualization of surface area of a sphere
- How do you prove this very different method for evaluating $\sum_{k=1}^nk^p$
- Sum of all natural numbers is 0?
- Stronger Nakayama's Lemma

- If $\operatorname{rank}(A)=m$, can we say anything about $\operatorname{rank}(AA^t)$?
- Let $V:= (C(),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.
- How to evaluate these indefinite integrals with $\sqrt{1+x^4}$?
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- Homotopy functions
- Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$
- Example for an ideal which is not flat (and explicit witness for this fact)
- Conjugacy Class in Symmetric Group
- Congruence Equation $3n^3+12n^2+13n+2\equiv0,\pmod{2\times3\times5}$
- How to solve simultaneous inequalities?
- Exponential growth of cow populations in Minecraft
- I don't understand what a Pythagorean closure of $\mathbb{Q}$ is; how are these definitions equivalent?
- Why does a time-homogeneous Markov process possess the Markov property?
- Proving $\sqrt2$ is irrational
- Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$