# Questions tagged [divergent-series]

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### Is there a superpolynomial sequence which is Abel-summable?

A sequence $a_n \in \mathbb{C}, \ n = 1, 2, 3, \dots$ is Abel-summable if for all $|x| < 1$ the sum $$g(x) = \sum_{n = 1}^{\infty} a_n x^n$$ converges and the limit $\lim_{x \to 1^{-}} g(x)$ exists....
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### Finding a variable P for which a sum converges [closed]

I need guidance in finding a variable P for which $\sum _{n=4}^{\infty }\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p$ converges, or proof that there doesn't exist such P ...
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### Theta-function in the lower half-plane

Standard theta function $$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$ has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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### Is regularization of infinite sums by analytic continuation unique?

There are ill-posed summations that we can assign values to, take for concreteness, $$S = \sum_{k=0}^\infty k$$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
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### Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$\sum_{p \text{ prime}} p$$ Neither of these questions obtained a ...
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### What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Working on an algebra of divergent integrals I came to the following relation: If $\tau=\int_0^\infty dx$ then $$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$ and this directly gives the following ...
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### Generalized limits

Cross-posted from Math SE. The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question: ...
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### Partitions of $\mathbb N$ which generate only divergent series of reciprocals of elements of those sets [closed]

This question is a result of some thinking about $\mathbb N$, divergent series and partitions of sets. Although elementary, I am not skilled enough to answer it at the present moment. Is it ...
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### Guessing of $n$th prime from “super- regularized” product of primes

( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.) We know "super-...
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### Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...
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### A suggestion for a superlimit

I have a question about summation methods. A value is assigned to a divergent sum. All methods agree that $\texttt{super-}\sum_{k=1}^{\infty} k^p = -\frac{B_{p+1}^{+}}{p+1}$ where $B_{p}$ are ...
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### Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here) ...
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### Why we cannot speak about the main or natural regularization?

Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it". But in practice if we consider some ...
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### What is the regularized sum of the following series (sum of all primes but spaced with zeros in place of non-primes)?

The sum over primes: $$\sum_{k=0}^\infty \{\text{k if k is prime, 0 otherwise}\}$$ I know that there is no known method to ascribe a reasonable value to the sum of the primes https://www.quora.com/...
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### Were there attempts to express derivatives of Delta function as polynomials of Delta function?

Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory? Particularly, it could ...
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### Hausdorff methods of summation

From the book of Boss "Classical and modern methods in summability": "The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
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### What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]

I tried Mathematica, various regularization methods, including Borel, with no result. On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at ...
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### Regularization on divergent series [closed]

I want to compute $$\sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}.$$ This series is surely not convergent for any ...
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### Is this a Borel summable $S = \sum_{k=0}^\infty (-1)^k (k!)a_k$ with $a_k$ alternating sequence?

let $S = \sum_{k=0}^\infty (-1)^k (k!)a_k$ a divergent series such that $b_k=(-1)^k (k!)a_k >0$ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
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### What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
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### Why is it customary to have formal series at infinity in the context of resurgence and 1-summability?

In the context of the theory of 1-summability and resurgence, it is customary to deal with formal series "at infinity" rather than at $0$ $$\sum_{n=0}^{\infty}a_nz^{-n}$$ This is stated for example ...
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### Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parallel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...
Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$? Experimentally this seems plausible (up through ...