# Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

14,567 questions
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### Continuous mapping theorem, multivariate case, joint distribution.

I came across the following problem. Convergence in the following always means weak convergence, i.e. $X_n \rightarrow X$ if and only if $Ef(X_n) \rightarrow Ef(X)$ for all $f$ bounded, continuous ...
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### Uniform convergence of polynomial approximation on Schwartz space

I have a question regarding uniform convergence of basis expansion in Schwartz space. For $L^2(\mathbb{R},\lambda)$, $\lambda$ Lebesgue measure, the partial sums of basis expansion (Hermite functions) ...
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### Banach space exercise

Suppose $X$ and $Y$ are Banach spaces and $T_{n} \in B(X,Y)$. If $T_{n}x_{n} \to 0$ in $Y$ for any choice of unit vectors $\{x_{n}\}$ in $X$, show that $\|T_{n}\| \to 0$.
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### pointwise convergence on interval and in $L^p$

I need to prove that sequence of functions $(f_n)_{n=1}^{\infty}$is pointwise convergent on $[0,1]$ but it is not convergent in the space $L_2[0,1]$ If I showed that $f_n \to 0$ on $[0,1]$ is ...
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### Show: if $\sum_{n>0} f(n)$ is convergent, then $\sum_{n>0} n^{1/n}f(n)$ is convergent [on hold]

If $\sum_{n>0} f(n)$ is convergent, then show that $\sum_{n>0} n^{1/n}f(n)$ is convergent . I am trying using Abel's test , but I can't find my way .
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### Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!}$ converge?

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!}$ converge? I have no idea how to do this. I have tried to use any trick I am aware of but can't figure this out. Can anyone help ...
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### Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
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### $\{f_n\}$ sequence of holomorphic functions converges uniformly.

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of holomorphic functions $f_n:U\to \mathbb{C}$. Suppose this sequence is Cauchy with respect to the $L^2$ norm. Then $\{f_n\}$...
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### Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.2 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging ...
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### How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
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### Convergence rates for SQP solver on strongly convex problem

Given the following optimization problem: $\boldsymbol{x}^* = \arg \min_{ \boldsymbol{x}} g(\boldsymbol{x})\\ s.t. : \boldsymbol{b} = A \boldsymbol{x} \\$ Where $\boldsymbol{A}$ is full row-...
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### Probability limit of log chi sqaured

I have the following: $$\Pr\left(T \ln(1+\chi_{1})>\ln(T)\right) \rightarrow 0$$ where $\chi_1$ is a chi-squared distribution with 1 degree of freedom and $T$ is an arbitrary real number and the ...
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### A question about the proof of Theorem 5.21 in Van der Vaar(1998)

If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\...
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### If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
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### Convergence of $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$

I was trying to find if the series $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$ converges or diverges. But I couldn't solve the question and I looked at the solution in here. In that page, Limit ...
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### Does $\sum_{i=2}^n \frac{3}{n\ln(n)}$ converge or diverge? [on hold]

I came upon this question while working: $$\sum_{n=2}^\infty \frac{3}{n\ln(n)}$$ And I was wondering whether it converges or diverges? A help would be greatly appreciated ! Thank you!
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### series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...