# Questions tagged [divergent-series]

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### Interchange summation order in the limit of number of elements going to $\infty$

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
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### A proposition for summing divergent series, but how should partial summation be defined at non-natural values?

Introduction I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
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### Divergent series summation beyond natural boundaries

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
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### Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$

I'm hoping to find a reasonable value to assign to the divergent series $\sum_{n=0}^\infty (-1)^n n^n$ and $\sum_{n=0}^\infty (-1)^n (xn)^n$. For the first one, I have obtained something around 0.71, ...
1 vote
156 views

### List of assigned values of divergent series

I'm hoping to find a list of divergent sums where the assigned value is generally accepted. For instance $\sum_{n=0}^\infty (-1)^n$ is generally accepted to be $\frac{1}{2}$. Moreover, its agreed upon ...
156 views

### 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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### Some exercise on the regularity of a summability method

I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
205 views

### Is there an asymptotic bound between converging and diverging series? [closed]

Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$, $$\log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}.$$ It is well known, ...
602 views

### Do smooth cutoff functions analytically continue functions?

My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
1 vote
80 views

### What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...