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### how to evaluate the following double summation to infinity without using integration method?

The expression is as follows: $\sum_{x=0}^{\infty}\sum_{y=0}^{\infty} \exp(-\sqrt{x^2+y^2})$ I have thought about using Taylor approximation to get started but it doesn't seem to get me anywhere. ...
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### Probability of sub-sequence of exact length to occur

Let's suppose that I have a sequence of length $L$ of uniformly distributed random numbers on interval $(a,b)$. How can I calculate probability that increasing sub-sequence of length $M,M <L,$ ...
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### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...
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### Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
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### Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
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### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...
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### Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...
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### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...
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### Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas: 1) The ...
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### Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$. We denote the least integer not covered by the arithmetic ...
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### Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
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### Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
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Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form $$A_{n}=B_{n}+C_{n}A_{... 4answers 1k views ### How to calculate the infinite sum of this double series? I'm calculating this double sum:$$ \sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2} $$I know the answer is$$ \frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16} $$which can be ... 4answers 2k views ### Why do Pell equations appear in Ramanujan's pi formulas? While answering this MSE question about the Pell equation x^2-29y^2=1, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit \... 7answers 3k views ### What problem would you base your mathcoin on? Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple (p,p+4,p+... 1answer 282 views ### Does this function have any exponential growth? Has anyone seen any function of the following type?$$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$The question is whether for some constant c>... 1answer 391 views ### The closed form of \sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n) The following series I'm interested in$$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$where \psi(n) is digamma function arose in the evaluation of an integral I posted on MSE, http://... 1answer 499 views ### A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm Let f_n \in L^2[0,1] be an orthonormal sequence and let c_n \in \mathbb C be such that \sum_{n = 1}^{\infty} |c_n|^2 < \infty. Does this imply that the sequence \sum_{n = 1}^{\infty}c_nf_n ... 3answers 489 views ### Asymptotic formulas for Monster-related modular functions? Define the following,$$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}j_{2A}(\tau) =\Big(\big(...
Please allow me to resort once again to the expertise of the MathOverflow community : During research I encoutered the following infinite series : \sum_{n=-\infty}^{+\infty} \frac{u^{2n}}{1+\rho^{...