Convergence of sequences and different modes of convergence.

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Finding the $p,\ r,\ q$ for which the series converges

I'm dealing with the series: $$\sum_{n=3}^{\infty} \frac{1}{n^p(\ln n)^q(\ln(\ln n))^r},$$ looking for the set of all $p,q,r$ such that the series converges. Is there a way to determine this without ...
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1answer
77 views

Does $\sum_{n=1}^\infty\sin(n)\sin\left(\frac{\pi}{2n}\right)$ converge?

I must determine whether if the following series converges, converges absolutely, or diverges: $$\sum_{n=1}^\infty\sin(n)\sin\left(\frac{\pi}{2n}\right)$$ By the comparison test, I have already found ...
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1answer
96 views

When does $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ absolutely converge?

Let $p>0$. I must find the values of $p$ for which the following series converges: $$\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$$ I have already successfully proven the following estimate by induction: ...
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2answers
22 views

Is this a valid step in a convergence proof?

I'm asked to say what the following limit is, and then prove it using the definition of convergence. $\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$. Is it valid to say that the sequence ...
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38 views

Finding convergence parameter

Find the value of the parameter $p$ for which the following series converge. Series of: $\frac{1}{(ln(k))^p}$ from $2$ to $\infty$.
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92 views

Simple convergence proof

I'm asked to prove, using the definition of convergence, that limits approach a certain value. For example, $$\lim_{n\rightarrow\infty}\dfrac{n^2+4}{n^2}.$$ I can see that it converges to $1$, but ...
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0answers
47 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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35 views

Convergence of Sequences Proof

Let $(x_n)$ and $(y_n)$ be convergent sequences. Use the definition of convergence (no limit theorems!) to prove that the sequence $(3x_n2y_n)$ converges. I'm having trouble doing this using the ...
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48 views

Determining a general sequence is bounded and has a particular limit.

I'm having a hard time with the following problem. I seem to be going around in circles in trying to prove either. Some tips in finding a proof for both would be appreciated. For $n\ge 1$, let ...
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24 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
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On convergence of the series $\sum_{k=3}^{\infty}\binom{k-1}{2}p^3(1-p)^{k-3}$

Given a number $0 \le p \le 1$ calculate if: $\sum\limits_{k=3}^\infty \binom{k-1}{2} p^{3}(1-p)^{k-3} = 1 $ So: $\sum\limits_{k=3}^\infty \binom{k-1}{2} p^{3} (1-p)^{k-3} = \sum\limits_{k=3}^\infty ...
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Measure theory question on relationship between convergence of functions and convergence in $L_1$

Q/ Let $\{f_n\}$ be a sequence of measurable functions that converge to $f$ almost everywhere. Does $f_n$ converge in $L_1$? Justify your answer. How about the converse? (i.e if $\{f_n\}$ converges to ...
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1answer
10 views

Uniform convergence of decreasing functions on increasing finite sets

Let $f_k$ be a function from the finite set $S_k$ to the real interval [0,1], with $S_k\subseteq S_{k+1}$. Let also $S=\bigcup_{k\ge 1} S_k$ and assume that $S$ is the set of rational numbers in ...
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73 views

Prove $\{n^2\}_{n=1}^{\infty}$ is not convergent

Definition of convergent sequence: There exists an $x$ such that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, $d(x_n, x) < \epsilon$. So the negation is that ...
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1answer
21 views

Requirements to fully prove convergence using the integral test.

Based on the theorem below, when using the integral test to prove the convergence or divergence of a series, does one need to also prove the series itself is decreasing, continuous and positive? Would ...
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22 views

how can we find such a counterexample for a nearly uniformly convergent sequence?

Pugh gives the definition of nearly uniformly convergence: $f_n:[a,b]\rightarrow \mathbb R$ converges nearly uniformly to $f$ as $n\rightarrow \infty$ if for each $\epsilon >0$ there is an ...
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1answer
39 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...
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53 views

completeness proof

I'm looking at this exercise solution and there is a last step which I do not really understand. Consider the set of continuous functions on the interval $X$, that is $C(X):=􏰁\{f:X→R \mid f \ ...
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1answer
42 views

Suppose $\sum_{n=1}^\infty b_n$ diverges, for $b_n > 0$. Show that the series $\sum_{n=1}^\infty \frac{b_n}{1+b_n}$ also diverges.

As the title says, given a series $b_n > 0$, where $\sum_{n=1}^\infty b_n$ is divergent: Show that the series $$\sum_{n=1}^\infty \frac{b_n}{1+b_n}$$ is also divergent. So I've defined the series ...
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2answers
30 views

Prove the limit converges $\epsilon$ proof

I believe there is an error in the solutions, and wanted to double check here. I need to show the following sequence converges to the proposed limit. $\lim \frac{1}{6n^2+1} = 0$ so we need to show ...
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3answers
360 views

Find limit for infinite sum

I'm trying to determine the limit of the sum $\lim_{n\to\infty} \sum\limits_{k=1}^n k^2/2^k$ Doing the convergence test shows the sum converges $\lim_{n\to\infty} \frac{(k+1)^2/2^{k+1}}{k^2/2^k} = ...
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Finding the limit of (3^n)/(n^3) and how to tell if it is convergent or divergent

In Calculus 2 we just started on doing sequences and I understand that to find the limit you can use l'hopital's rule and the sandwich theorem and a few other tricks but I'm generally confused on when ...
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42 views

Started this problem but can't finish it: Showing pointwise convergence for this summation

I know how to start this problem but am having trouble finishing the end of it. Any help would be great! Thanks We let $g_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ ...
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74 views

Convergence of $\sum_{m\text{ is composite}}\frac{1}{m}$

It can be easily show that the harmonic series $$\sum_{n=1}^{\infty}\dfrac{1}{n}$$ is divergent. Also it has shown that the infinite series of reciprocals of primes $$\sum_{p\text{ is ...
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1answer
45 views

Find the radius of convergence of complex power series

Give that the radius of convergence of $ \sum\limits_{n=1}^\infty a_nz^{n}$ is $R$, find the radius of convergence $R_1$ and $R_2$ of the following series: $$\sum\limits_{n=1}^\infty ...
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1answer
60 views

An infinite sum based on the mod-parity of Euler's totient function

Let $\bmod( m,k )$ be the remainder when $m$ is divided by $k$: $0,1,\ldots,m{-}1$. Let $\phi(n)$ be Euler's totient function: the number of relatively prime numbers smaller than $n$. So for ...
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1answer
177 views

A conjecture on uniform convergence of functions with a compact metric space

So I was having a discussion with a friend about this problem and we have conflicting views. Here it is We let $f_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ and we ...
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65 views

Convergence of $1/(x^2+y^2)^a$.

I'm curious about the convergence of the series: \begin{align} \sum_{x,y=1}^\infty \frac{1}{(x^2+y^2)^\alpha}\ ,\ \alpha \in \mathbb{N} \end{align} I'm wondering for what values of $\alpha$ this ...
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Does this series $\sum_{i=0}^n \frac{4}{3^n}$ diverge or converge?

I a newbie to series, and I have not done too much yet. I have an exercise where I have basically to say if some series are convergent or divergent. If convergent, determine (and prove) the sum of the ...
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1answer
38 views

If the sequence of distribution functions weakly converge, the sequence of corresponding subprobability measures converges weakly, too

Let $\mu,\mu_n$ be subprobability measures on $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$ $F,F_n$ be the distribution functions of $\mu,\mu_n$ with ...
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1answer
110 views

Is $\sum \sin(n^2)/n$ convergent?

Is the series $$\sum_{n\ge 1} \frac{\sin(n^2)}{n}$$ convergent? My thoughts so far: 1)This is an alternating series so the integration test does not work here. 2)The Weyl inequality roughly says ...
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The Convergence of an alternating series test

Can I confirm that $$\sum \frac{(-5)^{n}}{n^{3}}$$ converges by the alternating series test?
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19 views

How can I show this series converges uniformly?

Suppose $f(x) =\sum_{k=1}^\infty \dfrac{1}{k}\sin\left( \dfrac{x}{k+1}\right)$. Show that $f(x)$ converges uniformly on any closed, bounded interval $[a,b]$. I used the Weierstrass M-Test, but I am ...
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25 views

Does a terminating recurrence relation diverge?

Given the recurrence relation $$u_1=-3.25 \ \& \ u_{k+1}=\frac{4}{u_k+2}$$ is $\{u_k\}$ convergent? A quick check for the definition of convergence gives the following: If $\forall \epsilon \ ...
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38 views

Find Radius of Convergence of $\sum_{n=0}^\infty \frac{1}{2^n}\left(x-\pi\right)^n$

This is not a homework problem (I'm on break, so time for my own studies). Find the radius of convergence of \begin{align} \sum_{n=0}^\infty \frac{1}{2^n}\left(x-\pi\right)^n. \end{align} I have ...
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60 views

proving point wise convergence but no uniform convergence on f

Let f$_n$: E → $R$ be continuous functions for 1 ≤ n ≤ N. Let a$_k$$^n$ be N convergent sequences of numbers and assume $\lim_{k \to inf}$ a$_k$$^n$ = a$_n$. Let f = $\sum_{n=1}^N$a$_n$f$_n$. I am ...
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1answer
27 views

If sequence of r.vs $(X_n)$ is independent, then complete convergence is equivalent to convergence a.s?

We say that $X_1,X_2,.... $ is a completely convergent sequence to $X$ if $$ \sum_{n=1}^{\infty} P( |X_n -X | > \epsilon ) < \infty \; \; \; \; for \; \; each \; \; \epsilon >0$$ Question: ...
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1answer
27 views

Convergence of the supremum of random variables

I've a question regarding the convergence of the supremum of random variables. Assume $X_1, X_2, ...$ are i.i.d. and positive with $\mathbb{E}[X_i^4] < \infty$. Does $\sup\limits_{i=1, ..., n} ...
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1answer
51 views

Can the sum of reciprocals of a set without density converge? [duplicate]

It is known that if a set of natural numbers has positive asymptotic density then the sum of the reciprocals of those elements diverge. Let $\{a_n\}$ be an increasing sequence of natural numbers where ...
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84 views

How can I prove the following sequence is convergence [closed]

We know the floor function $[x]$ such that $[2.4]=2$ , $[-2.4]=-3$ , $[2]=2$ . How can I prove the following sequence $$x_n=\frac{[2^n \sqrt2 ]}{2^n}$$ converges to $\sqrt2$ . And in general the ...
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Convergence of a Sequence to a real number in [-1,1] [closed]

How to Prove that if $(u_n)$ is a sequence of real numbers, then there exists a subsequence $(u_n)_k$ , $k \in N$ such that $sin(u_n)_k$ , $k\in N$ converges to a real number in $[-1, 1]$. Please ...
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38 views

Limit of a sequence question

I have the following question in my assignment which I couldn't solve: Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $ I have to prove if the ...
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3answers
43 views

Uniformly convergent implies equicontinuous

I'm trying to prove that if I have a sequence of continuously differentiable functions $f_n$ that converge uniformly on $[a,b]$, then $\{f_n\}$ is equicontinuous. My idea is to use uniform ...
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1answer
29 views

An exercise from Chung on $L^1$ convergence

One of the exercises says: "Suppose $X_n \uparrow X$ almost surely, that each $X_n$ is integrable, and that $\sup \mathbb{E} (X_n) < \infty$. Show that $X_n \rightarrow X$ in the $L^1$ sense." ...
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1answer
23 views

Tail bounds of reciprocals

Suppose one knows that for a random function $f(n)$, $f(n)-a$ decays at some rate given by: $$Pr(|f(n)-t|>\epsilon)=g(\epsilon),$$for $g(\epsilon)\to0$, all as $n\to\infty$. If the above holds, ...
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47 views

How do you prove the almost sure convergence is not (in general) metrizable?

How do you prove the almost sure convergence is not (in general) metrizable? Many thanks for your help.
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67 views

Prove integral equals 1 in analysis

Prove that $$\lim_{n\to \infty} \int_0^1 e^{-x}\left(1+\frac{x}{n}\right)^n \text{d}x=1$$ You are free to use the fact that for each $x\in[0,1]$ the sequence $\{(1+x/n)^n\}$ is monotone increasing ...
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24 views

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous.

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous. Prove that if $(y_n)$ is a convergent sequence in $X'$ ...
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1answer
17 views

Converge almost surely

Consider the sequence of random variable$X_1,X_2...$ given by $$X_n= \exp(-n^2(Z-\frac{1}{n})), n=1,2...$$ where $Z$ is uniformly ditributed random variable on the interval$[0,1]$. Does this sequence ...
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1answer
20 views

Convergence of a product of sequences convergent in mean when one of them is bounded

Suppose $X_n\to X$ in $L^1$ and $V_n\to V$ in $L^1$ and $(V_n)$ is a bounded sequence. I'm trying to show that then $\mathbb{E}X_nV_n\to \mathbb{E}XV$. One has for all $N\in\mathbb{N}$ ...