# Questions tagged [divergent-series]

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### Can we just use the linear term of exponential sums to sum divergent series

Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n$ You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
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### 1+2+3+4+… and −⅛

Is there some deeper meaning to the following derivation (or rather one-parameter family of derivations) associating the divergent series $1+2+3+4+…$ with the value $-\frac 1 8$ (as opposed to the ...
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### Extending reals with logarithm of zero: properties and reference request

If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...
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### Generalizations of summation methods of divergence series

If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
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So I was considering the divergent everywhere but 0 power series $$f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n$$ Now one can do the following "questionable" manipulation $$f(x) = \sum_{n=0}^{\... 5 votes 3 answers 284 views ### Evaluating the series \sum_{n=0}^{\infty} n! x^n and inverse variable-fractional-derivatives So I was interested in formally assigning values to the completely divergent series G(x) = \sum_{n=0}^{\infty} n!x^n . I guess the question COULD end here if you already have an idea of how to ... 4 votes 2 answers 345 views ### Borel summation and the Abel function of e^z-1 This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,... 6 votes 0 answers 286 views ### Is there any intuition of why the both, regularized logarithm of zero is -\gamma and the regularized logarithm of Bernoulli umbra is -\gamma? If we take the MacLaurin series for \ln(x+1) and evaluate it at x=-1, we will get the Harmonic series with the opposite sign: -\sum_{k=1}^\infty \frac1x. Since the regularized sum of the ... 1 vote 0 answers 85 views ### Term-wise expectation of the Taylor series for 1/X yields asymptotic expansion for \mathsf EX^{-1}. What are the conditions? Migrated from the MSE. Let X\sim F_X denote a continuous random variable. Computing the first negative moment \mathsf EX^{-1} (assuming it exists) may not be tractable and thus a common tactic is ... 1 vote 2 answers 215 views ### What's the true regularized value of product of all natural numbers? Muñoz Garcia and Pérez-Marco - The product over all primes is 4\pi^2 claims that the regularized value of product \prod_{k=1}^\infty k is \sqrt{2\pi} and of \prod_{k=1}^\infty p_k over primes ... 1 vote 1 answer 851 views ### 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 ... 1 vote 1 answer 144 views ### Solving (or approximating) a certain delay differential equation I'm interested in finding the (unique?) solution to the set of delay differential equations$$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), f_x(w,x) = wf(w,w^2x)$$With the initial condition f(1,x) = e^... 4 votes 1 answer 143 views ### Reference request: Rigorously solving ODEs using divergent asymptotic series In my research I have come across a divergent asymptotic series \sum_{n =0}^\infty a_n f_n(x) that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond ... 7 votes 2 answers 451 views ### Adrastus, Proclus, and 2+8+50+288+… vs. 1+9+49+289+… According to the MacTutor essay "D'Arcy Thompson on Greek irrationals" (which I take to be a version of Thompson's original essay whose only liberty with the original text is giving English ... 2 votes 1 answer 232 views ### Linear combinations of geometric series Consider the uncountable-dimensional vector space V consisting of finite linear combinations of infinite sequences of the form (1,z,z^2,z^3,\dots) with z \neq 1 in \mathbb{C}. Since the ... 2 votes 0 answers 90 views ### Equality of bivariate formal series Is it possible to prove algebraically that the two series uniquely defined by the following equations are equal: L_1=uz+zL_1^2+z \partial_uL_1 and L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2 2 votes 0 answers 169 views ### Evaluating \sum_{n=0}^\infty n^k n! in p-adics, and its connection to the summation of divergent series Often, in the discussion of the regularization of the geometric series it is mentioned that \sum_{n=0}^\infty p^n converges in the p-adics, and indeed, that it converges to \frac{1}{1-p}. I had ... 3 votes 0 answers 142 views ### Arithmetic properties of error terms in divergent series Most people know the famous equation \sum_{k=1}^{\infty} k = -\frac{1}{12}, justified for example by interpreting the LHS as \zeta(-1). My question: does the sequence \{\frac{1}{12}+\sum_{k=1}^n ... 3 votes 2 answers 385 views ### 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 ... 11 votes 1 answer 909 views ### 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 ... 8 votes 2 answers 1k views ### 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, ... 2 votes 1 answer 212 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 ... 4 votes 1 answer 179 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$$converges and the limit \lim_{x \to 1^{-}} g(x) exists.... 3 votes 0 answers 53 views ### Some exercise on the regularity of a summability method I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ... 3 votes 1 answer 229 views ### Is there an asymptotic bound between converging and diverging series? [closed] Let us define for every k\in\mathbb{N} and every large enough x\in \mathbb{R},$$ \log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}. $$It is well known, ... 6 votes 0 answers 1k views ### Do smooth cutoff functions analytically continue functions? My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ... 1 vote 0 answers 91 views ### What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have? I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ... 2 votes 0 answers 211 views ### Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"? There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ... 4 votes 0 answers 255 views ### Is there a conjecture about the bounds (constant or a function) of \sum_{n \le x} \mu(n)/\sqrt{n} Here \mu(n) is Möbius function and M(x) is Mertens function. The computations show that the partial sums \sum_{n \le x} \frac{\mu(n)}{\sqrt{n}} stays between -0.2 and -1.2 when 10^1<x<... 3 votes 0 answers 406 views ### What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed] I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities.$$\...
We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum_{n\in A} n^{-1}$ ...