Convergence of sequences and different modes of convergence.

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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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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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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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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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41 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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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
44 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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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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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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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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132 views

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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2answers
45 views

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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1answer
24 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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0answers
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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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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26 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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37 views

Convergence of $\sum_{i \in \mathbb{N}} i \cdot a_i = 0$ given a convergent integer sequence $a_i$

Consider a sequence of integers $a_i \subset \mathbb{Z}$ which is identically equal to $0$ for all $i > I > 0$, there is at least one $a_i \neq 0$ such that $1 \leq i \leq I$ and $$\sum_{i \in ...
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1answer
83 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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50 views

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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37 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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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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28 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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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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66 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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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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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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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 $N\in\mathbb{N}$ ...
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33 views

Sequences and series of $\tan^n x$

Please help me with this question: Investigate the convergence of the sequence $$\tan x, \quad \tan^2 x, \quad \tan^3 x, \quad \dots, \quad \tan^n x$$ for $x \in (-90^\circ, 90^\circ)$. ...
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If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy.

If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy. Proof: $|a_{m_1}-a_{n_1}|\lt \epsilon_1$ and $|b_{m_2} - b_{n_2}|\lt \epsilon_2$ Then take ...
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42 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
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Ratio Test Interesting Issue

I was hoping someone could help me with this interesting situation that came up while I was teaching intervals of convergence today using the ratio test. The problem asked to find the radius and ...
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38 views

lim inf sup integral

I have a general question about integrals of sequence of functions. Suppose $f_n \rightarrow f$ pointwise. Can I automatically say that $\lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty}inf ...
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Convergence epsilon - check my proof please

Fairly straightforward question (I hope so anyway), would be very grateful if someone could check my proof. I need to show that $\frac{n+1}{n+3} \to 1 \space \text{as} \space n \to \infty$. I start ...
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When is the convolution of a product the product of convolutions?

Although the convolution of the product is not the product of the convolution, i.e. $$fg*h\neq (f*h)(g*h).$$ I am wondering if this true (for a suitable class of functions) in the limit when one ...
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26 views

When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
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Does $\sum_{n\in\mathbb{N}}a_n<\infty$ imply $\sum_{m\in\mathbb{N}}\sum_{n=1}^ma_n<\infty$

Let $(a_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ and $$\sum_{n\in\mathbb{N}}a_n<\infty\tag{1}$$ Can we deduce $$\sum_{m\in\mathbb{N}}\underbrace{\sum_{n=1}^ma_n}_{=:b_m}<\infty\;?$$ $(1)$ implies ...
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Convergence/divergence of geometric series when $k = 1$?

Usually, when we try to determine convergence/divergence, we simply find the quotient $k$, and if $|k|<1$ we say the series is convergent. At least this is how the textbooks present it. In set ...
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Proving a function has a convergent subsequence

Prove that $Y_n = tan(n)$ has a convergent subsequence Thanks for any help provided Not asking for direct answer, but strong hints/the solution would be great.
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497 views

Counter-example for misused theorem in series convergence test

I had a math exam not so long ago and got my result back, I'm happy with the result but there is a question for which my teacher gave me an explanation (for me not having the points) but I still think ...
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1answer
46 views

$\sum_{n=1}^{\infty}\frac{x^n}{n}$ converges for $|x|<1$

I wish to show $\sum_{n=1}^{\infty}\frac{x^n}{n}$ converges for $|x|<1$, preferably using the comparison test. I have done it using the limit ratio test, then having to show that ...
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2answers
48 views

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge. My proof : We know that $\{a_n\}$ converge therefore : $$\lim_{n \to \infty} a_n = L$$ All $\epsilon>0$ exist $N \in \mathbb{N}$ so ...
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1answer
33 views

Convergence of $f_n(x)=n^2f(nx)$ in the sense of distributionas

Let $f$ be a test function such that $\int_{-\infty}^\infty f(x)dx=0$ and $f_n(x)=n^2f(nx)$. Find the distributional limit $\lim_{n\to\infty}f_n$. How can I use the Dominated Convergence Theorem ...
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27 views

Divergence in probability

The sequence $(X_n)$ is said to diverge to $+\infty$ in probability if $\mathbb{P}\{X_n>b\}\to 1$ as $n\to\infty$ for every $b\in\mathbb{R}_+$. If $(X_n)$ diverges to $+\infty$ in probability and ...