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        Questions tagged [limits-and-convergence]

        Convergence of series, sequences and functions and different modes of convergence.

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        A question on a double series for $\frac{-\pi}{2}\left(\zeta(3)-1\right)$, deduced from the generating function $\sum_{k=0}^\infty\zeta(2k)z^{2k}$

        Using (25.8.6) from the Digital Library of Mathematical Functions (in this section 25.8) I wrote a statement with the help of Wolfram Alpha online calculator. I got $$\sum_{k=0}^\infty\sum_{n=1}^\...
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        51 views

        Effect of repeated subtraction of the average of average function values in coordinate directions

        Questions: assuming $$a\lt b,\ c\lt d;\ \ (x,y)\in [a,b]\times[c,d];\ \ f_0: (x,y)\mapsto z\in\mathbb{R};\ \ |a|,\ |b|,\ |c|,\ |d|,\ |z|\lt\infty$$ $$0\quad\lt\quad\left|\int_a^b{f_0(x,y)dx}\...
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        Compactness lemma: approximate sequence in the space X and the limit not in the same space

        Often, when we try to solve some PDE problem, we construct first a sequence of approximate solutions. To construct an exact solution we need to show that a sequence (or at least some subsequence) of ...
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        Prove an existing formula for a limit of a specific sum

        Prove that$$\lim_{n\to\infty}\frac1n\sum_{i_1,i_2,...i_k=1}^n\lambda_1^{|i_1-i_2-s_1|}\lambda_2^{|i_2-i_3-s_2|}...\lambda_k^{|i_k-i_1-s_k|}$$is equal to$$\sum_{j=1}^k\lambda_j^{S+k-1}\prod_{l=1,l\ne j}...
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        Sufficient conditions for taking limits in stochastic partial differential problems

        Let's say we have a (parabolic) Cauchy problem: $$ (1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$ $$(2) \hspace{0.5cm} u(x,0)=u_0(x), $$ ...
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        114 views

        What is the current fastest method to calculate Lerch's Phi transcendent?

        Lerch's Phi transcendent is $$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$ I am trying to compute this for the following parameters: $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...
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        Is there any statistically convergent real sequence, which is not almost convergent? [migrated]

        I have read that almost convergence and statistical convergence are incompatible (i.e. not comparable). For this both of below must be satisfied : There exists a statistically convergent real ...
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        87 views

        Limiting distribution of “scatter matrix” $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

        Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s. ...
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        32 views

        Limits of the wave equation with piecewise constant propagation speed

        This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty. Consider a wave equation $$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...
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        Largest eigenvalue scaling in a certain Kac-Murdoch-Szegö matrix

        I'm looking at $N\times N$ matrices $M_N$ with elements $$M_N=\left( \rho^{|i-j|} \right)_{i,j=1}^N,$$ where $\rho$ is a complex number of unit modulus. These matrices with $\rho\in\mathbb R$ and $|\...
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        28 views

        Extension of a “statistical limit functional”

        $c=$The set of all real convergent sequences $l_\infty=$The set of all real bounded sequences Clearly $c\subset l_\infty$ $f:c\to \mathbb R$ is called limit functional defined by $f(x)=\lim\limits_{...
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        Find limit of sequence defined by sum of previous terms and harmonics

        I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
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        Convergence acceleration of a series by using optimal parameters

        One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
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        117 views

        Comparison of several topologies for probability measures

        Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
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        Linear programming with a convergent coefficient

        The following linear programming problem $x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$ has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...

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