Convergence of sequences and different modes of convergence.

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Convergence parameter: Find the value of $p>0$ for which the series converge

For the sum for $k=2$ to infinity: $$\frac{\ln k}{k^p}\ $$ The textbook says the answer is $p>1$.
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0answers
21 views

Radius of Convergence of a Complex Taylor Series

I've recently been doing some complex analysis questions and come across a few of this type: Find the radius of convergence of the Taylor series at $z=-1$ of the function ...
2
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1answer
19 views

Convergence almost everywhere

Let consider rational numbers $\{r_n\}_{n=1}^{\infty}$ on [0, 1]. How to prove, that such sum $$\sum_{n=1}^{\infty}\frac{1}{n^2|x-r_n|^{0.5}}$$ converges almost everywhere on [0, 1]. There are my ...
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1answer
1k views

Uniform convergence of series $\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$

Using Dirichlet series test I've proves that series $\displaystyle\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$ converges for all $x\in\mathbb{R}$. How to determine whether the series ...
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0answers
11 views

Inequality/ Convergence for two operators with functional calculus

Given a sequence of functions $f_n \to f$ in $L^\infty(\mathbb{R}^2)$ and two self-adjoint, unbounded operators $A, B$ is it true that $\|f_n(A,B) - f(A,B)\| \to 0$? With only one operator I can ...
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2answers
24 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
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1answer
36 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
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1answer
32 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...
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1answer
40 views

Identify the radius of convergence from ordinary generating function

Assume we have the ordinary generating function $f(x)$ of a series: $f(x) = \tan x$ Can we identify the radius of convergence for this series?
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164 views
+100

Weak and Probability convergences

I have a question about this page, from Topics in Random Matrices Theory, of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
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1answer
50 views

Suppose that $\sum a_i$ converges and that $a_i\geq0$ for all 𝑖.

Suppose that $\sum a_n$ converges and that $a_n\geq0$ for all $n$. For each $n$, let $e_n=\pm1$. Then, prove that $\sum e_na_n$ converges. Can I simply say that ∑|eᵢaᵢ| = ∑aᵢ so that ∑eᵢaᵢ converges ...
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20 views

Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
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1answer
12 views

Series convergence or divergence

Determine whether this series converges: $\Sigma e^{-j+sinj}$ I know that this series is $\leq$ than $\Sigma e^{-j+1}$, but I am having trouble getting this in a form appropriate for a convergence ...
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1answer
47 views

Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle ...
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1answer
27 views

Convergence of $\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$

$$\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$$ where $p>0$ And $n\in\mathbb{N}$. I understand we can compare this to $$\int_{0}^{\pi} \frac{1}{(x+n\pi)^{p}} dx$$ which tells us it converges ...
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2answers
116 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
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1answer
46 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
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0answers
16 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
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1answer
20 views

$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges [duplicate]

Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges. I have looked at an answer here, but I do not understand what these results give us. For example, in the first answer: ...
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3answers
110 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
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1answer
37 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
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1answer
54 views

Identify the radius of convergence [on hold]

For the ordinary generating function $f(z) = \frac{z^3 +1 }{z^3 -1}$, how can we identify its radius of convergence? And is this function meromorphic?
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3answers
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Product of two power series

Say if I define a power series over some arbitrary field $F$ as $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$ Then can I say: $$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
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2answers
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Need help to understand calculating fourier transform of 1

this issue is related to a physics problem, but since it is mathematical I will post it here. When calculating the following Fourier transform $$ -i\int_\infty^\infty dt~ e^{i\omega ...
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34 views

Bound on $f_n'$ implies uniform convergence of $f_n$?

Let $f_n$ be a sequence of functions that converge pointwise to a function $f$. Suppose I know that $|f_n'(x)| \leq C(x)$ where the constant doesn't depend on $n$. How do I conclude that $f_n \to f$ ...
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1answer
46 views

Convergence in probability implies convergence in mean under one additional condition

Prove that if random variables $X_n$ are dominated by an integrable random variable then $E[X_n] \to E[X]$ follows if $X_n$ converges to $X$ in probability. Hint: Use the following theorem : A ...
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1answer
52 views

Convergence of series with sum

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact convex set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i ...
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2answers
931 views

Does almost everywhere convergence imply convergence in $L^p$?

Is it true that if $\Omega$ is an open, bounded subset of $R^{N}$, $u_{n} \to u$ almost everywhere, $1<p<\infty$, then $\|u_{n} - u\|_{L^{p}} \to 0$? Here sequence and the function are in ...
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2answers
50 views

Does differentiability imply convergence

Can we say that if the limit of a sequence of functions is differentiable then the sequence is convergent? I mean, I know that $\frac{\partial f(x,t)}{\partial x}$ exists. If I specify a sequence ...
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3answers
28 views

Convergence of an Alternating Series

For this question, I`m trying to determine the values of $p$ in which the series converges. The series is: $\sum_{n=1}^\infty \frac{(-1)^n}{n^p}$. I already know (by looking at the answer to the ...
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3answers
189 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$

I'm trying to prove the convergence of $$ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}$$ with $\alpha > 1$. For $\alpha \geq 2$ I can use the comparison test ($\sum_{n=1}^{\infty} \frac{1}{n^2}$ ...
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0answers
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Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0<p<1$? I have the ...
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1answer
23 views

Is this set measurable? (Set of points where a sequence converges)

Let $M$ be a manifold. Suppose that $u_n:M \to \mathbb{R}$ are measurable and we have $u_n(s) \to u$ a.e. in $M$. Does it follow that the set $A=\{s \in M : u_n(s) \to u(s)\}$ and $A^c$ are ...
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An example of a sequence of continuous real functions pointwise convergent, but nowhere locally uniformly convergent? [duplicate]

I've been trying to come up with an example of a sequence of continuous real function which would converge pointwise everywhere, but nowhere converge locally uniformly, but I can't really think of ...
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2answers
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How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
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0answers
35 views

integral of a sequence

Let $[a,b]$ be any closed interval in $\mathbb{R}$ such that $0\notin [a,b]$ and $f_n\in L^2(\mathbb{R})$ for $n\geq0$. If $\int_a^b f_n\phi\rightarrow \int_a^b f_0\phi$ for every $\phi$ in Schwartz ...
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3answers
74 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
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2answers
62 views

Is the sequence $(x_n)$ convergent in the space $L_1(0,1)$

Is the sequence $(x_n)$ convergent in the space $L^1(0,1)$ ? $x_n(t)= n^2 t^n (1-t^2)$ for $n\in\mathbb{N}$. norm: $\|x\|=\int_{(0,1)} \left|x(t)\right| \; dt$ I think it should ...
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2answers
28 views

Convergence of an improper integral(with parameters)

I'm trying to find solution to this problem: For what pairs (a; b) of positive real numbers does the improper integral $$ ...
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1answer
54 views

Prove that $\varphi^n(t) \rightarrow 0$ when $n \rightarrow \infty$

I have to prove a lemma: If $\varphi: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ is monotone increasing and $\varphi(t) < t, \ \forall t \in \mathbb{R}_+$, then $\varphi^n(t) \rightarrow 0$, $(n ...
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1answer
24 views

Prove Uniform Convergence of Series of functions-Help?

Let F0 be a bounded Riemann integrable function on [0, 1]. For n ∈ N, define $F_n(x)$ on [0,1] by $F_n(x)$ = $\int_{0}^{x}$ $F_{n-1}(t)$ dt 1) Prove that for all n∈ N and x∈ [0,1], we have ...
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4answers
127 views

Why do some series converge and others diverge? [on hold]

Why do some series converge and others diverge? What causes the divergence or convergence of a series and why does that cause such a behavior? For example, why does the harmonic series diverge, but ...
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1answer
32 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
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2answers
36 views

Testing A Series For Convergence

Determine whether the series $\sum_{n=0}^{\infty} \frac{3n^2 + 2n + 1}{n^3 + 1}$ with n from 0 to infinity converges or diverges. So far I thought about dividing the numerator by the denominator, ...
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2answers
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Searching for a function

I'm searching for a funcion with this values: f(0)=0 f(1)=1 f(2)=3/2 f(3)=7/4 ... lim from x to infinity: f(x) = 2 I don't want the recursive way to define f.
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1answer
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Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
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1answer
29 views

Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
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1answer
26 views

Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
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1answer
48 views

Convergence in measure of integrable functions implies limit is integrable?

I'm going through my handwritten notes for my upcoming exam (so not homework) and the above was stated but not proven in class. The full statement is a little different, but the above part is the only ...