# Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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### Convergence of Sequences with respect to Ultrafilters on the Natural numbers in Compact Hausdorff Topological Spaces

Let $(X,\mathcal T)$ be a topological space. Let $\mathcal F$ be an ultrafilter on $\mathbb N$. We say that sequence $(x_n)_{n\in\mathbb N}$ converges with respect to $\mathcal F$ to $x$, iff for all ...
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• 61
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### Odd Function/Integral when evaluating a Moment

I am studying the Laplace (Double Exponential) distribution, and I have the following quote from Siegrist, which is quite direct, and not bothered about conditions being fulfilled: That the odd order ...
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### Measurability of sets used to define convergence in measure

In Folland's Real Analysis: Modern Techniques and Their Applications, the following definition is given for a sequence of functions converging in measure. We say that a sequence $\{f_n\}$ of ...
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### Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets)

To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.) Beginning of book quote End of book quote In the remark ...
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### Solving the non-linear recurrence $d_i + 2 i d_{i-1} = e_{i+1}$ (via generating functions)

I've been studying the recurrence $$d_i + 2i d_{i-1} = e_{i+1} =: (-2a)^{i+1}e^{-a^2} \quad\quad\quad a \in \mathbb{R}$$ attempting to solve for the sequence $(d_i)$. (The $d_0$ case is defined ...
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### Show that convergence of mean square implies convergence of mean

Let $\{y_n\}$ be a sequence of real numbers and let $$\bar{y}_n = \frac{1}{n} \sum_{i=1}^n y_i \qquad s_n =\frac{1}{n}\sum_{i = 1}^n y_i^2$$ Suppose there exists a real number $s$ for which s_n \to ...
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### Understanding convergence rate of gradient descent

I am currently learning about gradient descent. For the convex case, I found this estimation in Nesterovs book: $f(x_k)-f^* \leq \frac{2L\|x_0-x^*\|^2}{k+4}$ Nesterov doesn't use the big o notation ...
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### For what values of $k$ does the double integral of $\frac{1}{\|x_A - x_B\|^k}$ over intersecting surfaces converge?

I am working on a problem related to defining a potential energy function based on the distance between two intersecting surfaces in Euclidean space. Specifically, I am considering the following ...
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### Proof verification: Show that if a series is conditionally convergent, then the series from its positive terms is divergent.

Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent. Please, help me to ...
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