Convergence of sequences and different modes of convergence.

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4
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3answers
142 views

Is this a valid proof for ${x^{x^{x^{x^{x^{\dots}}}}}} = y$?

So I got this challenge from my teacher. Solve ${x^{x^{x^{x^{x^{\dots}}}}}} = y$ (eq. 1) for $x$. My attempt: As $x^{y^z}$ per definition equals $x^{y \cdot z}$, then $x^y = y$ from (eq. 1). ...
0
votes
2answers
26 views

Is limit function of continuous even functions always continuous?

Limit function of Uniform Continuous functions (pointwise converge) is always continuous. It is the fact. Limit function of continuous functions is not always continuous (am I right?) However, for ...
0
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0answers
13 views

Reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem.

I want a reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem. I understand some intuition behind taking independent copies of the random ...
0
votes
2answers
25 views

Prove absolute convergence from alternants

He failed to get the show in its entirety in this series, one I could indicate how working with this kind of series? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
0
votes
1answer
20 views

Convergence and Divergence and Using Various Methods

I am totally confused with the idea of convergence and divergence and which method to use to proof it. An example is a question like this: Does this integral converge? $$\int_{12}^\infty ...
2
votes
0answers
92 views

Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$

This is by far my most ambitious proof attempt to date and I'm not very good at them; so even if the proof is correct I would still appreciate feedback on nomenclature, clarity, elegance, etc... ...
0
votes
0answers
19 views

Converging quicker?

I am currently trying to find an expansion for an equation, however it converges rather slowly. It is in the form $$\sqrt{\frac{kx^2+1}{cx+1}}$$ where k,c > 1 and x < 1. Is there anyway around ...
0
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3answers
62 views

Use the ratio test to determine if the infinite series $\sum_1^∞ {n^4\over 4^n}$ converges or diverges [closed]

Use the ratio test to determine if the infinite series $\sum_1^∞ {n^4\over 4^n}$ converges or diverges
2
votes
2answers
72 views

Divergence of Unusual Series

I already prove this series. If $u_n > 0$ and $\frac{u_{n+1}}{u_n} \le 1 - \frac{1}{n} - \frac{\alpha}{n\log n}$, where $\alpha >1$, then $\sum u_n$ converges. However, I could not prove the ...
1
vote
1answer
12 views

Proving a.s. convergence by probabilistic convergence

Consider a sequence of random variable $\{X_n\}$. Let $$A_n = \sup\{|X_k - X_l|: k,l \geq n\}$$ $$B_n = \sup\{|X_k - X_n|: k \geq n\}$$ Now to prove a.s. convergence of $\{X_n\}$, I have seen in a ...
0
votes
0answers
17 views

Pointwise/uniform convergence and conditional/absolute convergence of series of functions

I'm trying to understand the relationships between pointwise/uniform convergence and conditional/absolute convergence of series of functions. Given that some series, say, converges conditionally, can ...
1
vote
1answer
17 views

Restatement of definitions of pointwise/uniform convergences

Consider the definition of pointwise convergence: sequence $f_{n}$ converges pointwise on A to f $$\forall x\in A\;\forall\epsilon>0\;\exists N:\; n\geq ...
3
votes
1answer
62 views

Finding Function's Extension and Its Unique Existence.

Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition: There is a sequence $ \epsilon_n \gt 0 $ ...
1
vote
1answer
19 views

Interval of convergence? (Relatively simple question)

What is the interval of convergence of the power series: $\dfrac{(-1)^{(n-1)}x^n}{n^3}$ I know it should be |x| < 1, but does that mean the interval of convergence is $(1,-1)$ or $(-1,1]$ or ...
0
votes
2answers
57 views

Convergence of series $n/(n+1)(n+2)(n+3)$

I have some problems to determine the sum of the following series, $$ \sum \limits^{\propto }_{n=1}\frac{n}{(n+1)(n+2)(n+3)} =\frac{1}{4} $$ appreciation or appreciate any help.
1
vote
2answers
88 views

Why $\zeta(-2) $ is not $\sum_{n=1}^{\infty}\frac{1}{n^{-2}}$? [duplicate]

Let $\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}$ a standard formula. I'm confused if you tell me: does this series: $\sum_{n=1}^{\infty}\frac{1} {n^{s}}$ converge? I will answer you: this series ...
1
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2answers
36 views

Absolute or Conditional Convergence of Summation of Double Log Sequence

$$\sum_{k=3}^\infty \frac {(-1)^k \log k}{k\log (\log k)} $$ Can you show me how to prove this? What test do I have to use? Double log is always confusing me. I have two more summation of ...
2
votes
2answers
45 views

When is the series converges?

Let the series $$\sum_{n=1}^\infty \frac{2^n \sin^n x}{n^2}$$. For $x\in (-\pi/2, \pi/2)$, when is the series converges? By the root-test: $$\sqrt[n]{a_n} = \sqrt[n]{\frac{2^n\sin^n x}{n^2}} = ...
1
vote
1answer
34 views

Cesàro means of conditionally convergent series

I am interested in the limit of Cesàro means (not sums or means of sums) of sequences whose corresponding series are conditionally convergent. By that I think I mean $\lim_{n \to \infty} \frac{1}{n} ...
2
votes
3answers
58 views

Examine the convergence of a sequence $\{a_{n}\}$ which is given by $a_{1}=a>0,a_{2}=b>0, a_{n+2}=\sqrt{a_{n+1}a_{n}},n\ge 1$

I used inequality between arithmetic and geometric means to show that a sequence $\{a_{n}\}$ is bounded: $$a_{n+2}=\sqrt{a_{n+1}a_{n}}\le \frac{a_{n+1}+a_{n}}{2}$$ Solving this, I get quadratic ...
1
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2answers
31 views

If $0 \leq f_n \leq g_n \rightarrow h$ in $L^2$ and $\int f_n^2 \rightarrow \int h^2$ then $f_n \rightarrow h$ in $L^2$

Let $(X,m)$ be a measure space, $(f_n)_n, (g_n)_n, h \in L^2(m)$. I would like to prove that if $0 \leq f_n \leq g_n$, $g_n \rightarrow h$ in $L^2$ and $\int f_n^2 \rightarrow \int h^2$ then $f_n ...
1
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1answer
28 views

inner product space problem $(x_n,y_n)\to 0$

If $(y_n)$ is a bounded sequence in an inner product space, and $(x_n)$ is a sequence converging to zero, prove that $(x_n,y_n)\to 0$. Where $(x_n,y_n)$ is the inner product. Since $(y_n)$ is bounded ...
2
votes
1answer
48 views

If $f_n \rightarrow f$ in $L^1$ then $\sqrt f_n \rightarrow \sqrt f$ in $L^2$

Let $(X,m)$ be a measure space, $(f_n)_n,f \in L^1(m)$ all non-negative and $f_n \rightarrow f$ in $L^1$. Is it true that then $\sqrt{f_n} \rightarrow \sqrt f$ in $L^2$? My try: $$ \int (\sqrt{f_n} ...
0
votes
1answer
27 views

Is there a counter-example to this problem on convergence of a bounded, strictly increasing sequence?

I came across this problem in an old set of class notes: Let $A \subset \mathbb{R}$ be a nonempty, bounded set. Let $\alpha = \sup{A}$, and let $(a_n)$ be a convergent sequence in $A$, with $a = ...
1
vote
4answers
51 views

Examine the convergence of a sequence ${a_{n}}$: $a_{n+1}=a_{n}-\sin(a_{n}),0\le a_{1}<\pi$

${a_{n}}$: $a_{n+1}=a_{n}-\sin(a_{n}),0\le a_{1}<\pi$ One way to do it is to show that the sequence is bounded and monotonous. How to show that it is bounded? If $$-1\le \sin(a_{n})\le 1$$ ...
0
votes
1answer
58 views

Prove the series converges, which sequence is defined inductively

Define $t_n$ inductively by $t_1=1$ and $\displaystyle t_{n+1}= \frac {t_{n}}{1+t_{n}^ \beta}$, where $\beta$ is fixed, $0 \le \beta \lt1$. Prove that $\sum_{k=1}^ \infty t_n$ converges. [Hint : Find ...
2
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0answers
32 views

Why is convergence of measures tested against functions?

This question is to help my intuition. Why do we test the convergence of measures against different classes of functions and not use definitions like: If $(B,\mathcal{B})$ is a measurable space then ...
2
votes
2answers
93 views

How to define the set $\lim_{n\to\infty}\{1/n,2/n,…,n/n\}$ rigorously?

How to define the set $\lim_{n\to\infty}\{1/n,2/n,...,n/n\}$ rigorously? For example, when asked to give a dense set in $[0,1]$, we let $S_n=\{1/n,2/n,...,n/n\}$ and take the union of all $S_n$ ...
0
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2answers
31 views

Show that for any numbers $p$ and $q$, $\{f\in C[a,b]:p\leq f(x)\leq q\}$ where $x\in [a,b]$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$.

Show that for any numbers $p$ and $q$, $\{f \in C[a,b] \mid \forall x\in [a,b]: p\leq f(x)\leq q\}$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$. We must show that if $f_n\to F$ and ...
1
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0answers
34 views

Help with whether series converges

how can you tell if $\sum_{n=1}^\infty \frac{(2n-1)!!^{1/5}}{(2n)!!^{1/5}}$ converges or not? I tried Raabe's test but didn't get anywhere. Thanks in advance.
0
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2answers
136 views

Limits and Series in Smooth Infinitesimal Analysis

I just learned a tiny bit about SIA. While it is interesting, that it handles derivatives so easily, I wonder: Can we still recover the concepts of limits (of sequences) and especially series, to ...
1
vote
1answer
63 views

Prove that the sequence is a weakly convergent sequence

I am trying to solve the following exercise: Fix $a,b \in \mathbb{R}^n, 0 < \lambda < 1.$ Define $$ u_k(x)= \left\{\begin{array}{ccc} a & \text{if } & \frac jk \leq x < ...
3
votes
5answers
86 views

Ratio Test and $\sum \frac{n^n}{4^n\cdot n!}$

I'm having trouble showing that the series $\sum_{n=1}^\infty \frac{n^n}{4^n\cdot n!}$ converges. The only hint I've been given is that the $\lim_{n\to\infty} (1 + 1/n)^n = e$. I've tried setting ...
2
votes
1answer
27 views

Prove that a family of mollifiers converges in $L_\text{loc}^\infty$

Let $B_1$ be the open ball with radius $1$ around $0\in\mathbb{R}^n$ and $\phi:\mathbb{R}^n\to [0,\infty)$ with $\phi\in C_0^\infty(B_1)$, i.e. $\phi$ is infinitely many differentiable in $B_1$ and ...
3
votes
1answer
31 views

Convergence of sum of a linear combination of Poisson variables

Let $Y_j$ with $j=1,...,m$ be independent Poisson random variables with parameter $\lambda_j$. I need some hints to find (provided that it exists, so with some condition on the sequence $\lambda_j$) ...
0
votes
3answers
99 views

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not. I tried to use comparison test or limit comparison test but could't find a suitable function. How can I determine what type ...
-1
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0answers
23 views

Is convergence defined only for infinite series? Also if it is then how is it defined for Binomial Series?

Also how is binomial series convergent for any value of x? Also we say that the binomial series is convergent for |x|<1, do we mean that the Taylor formulia for binomial series will not be valid if ...
0
votes
1answer
25 views

Finding limit of sequence with basic rules

I'm trying to find the limit of $(n^2-\sqrt{n^4-n^2+2})$ as $n$ approaches infinity with basic tools such as the laws of limits and L'Hôpital's rule. I'm pretty much stuck at the fact that the limit ...
2
votes
0answers
27 views

Under what conditions on the experiment does bootstrapping work?

For a proof I would like to pretend that the uniform distribution on a finite set of samples from a 'source' eventually becomes the source's distribution a.s. when you keep adding samples. I am not ...
3
votes
0answers
57 views

Prove that $f_n(x)$ is discontinuous at $x = 0$.

I am having problems with the following exercise, I am not sure if my procedure is correct. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} 0 ~~~if~~x = 0 ...
1
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1answer
19 views

On convergence theorems for limit calculations.

I came across the following limit $$\underset{n\rightarrow\infty}{lim}\int_{0}^{\infty}\frac{\sin{e^x}}{1+nx^2}dx$$ and I can't use any convergence theorem to show that this is equal to ...
4
votes
1answer
95 views

Converge of a sequence in $L^p(\mathbb{R}^3)$

Let $f(x)\in L^p(\mathbb{R}^3)$ for every $p\in [1, \infty]$. Let $B(n)\subset \mathbb{R}^3$ be the ball of radius $n$ centered at the origin. I want to show that the sequence ...
0
votes
1answer
39 views

Problem showing the probability of a sequence of sets converges to 1.

Im reading notes on M-estimators and the following Theorem is presented: Theorem: Assuming [...], the local M-estimator $\hat{\psi}_n$ will be well defined with probability converging to 1. ...
2
votes
0answers
32 views

Radius of Convergence of power series of Complex Analysis 111

I need to find radius of convergence of this complex power series $$\sum_{i=1}^\infty {e^{in-2in^2}+in \over n^3+3^{in}}{(z+1-i)}^n$$ I tried ratio test and also Cauchy Hadamard but nothing seems to ...
0
votes
2answers
47 views

Radius of convergence for fun complex sum!

I have dealt with radius of convergence for simple series, but this one is literally complex: $\frac{1}{1-z-z^2}=\sum_{n=0}^\infty c_nz^n$ How does one calculate the radius of convergence here? I ...
1
vote
2answers
49 views

Does Liebniz-Criteria become a necessary condition for convergence if a_n is monotonically decreasing?

The Leibniz Criterion says that if the sequence $a_n$ is monotonically decreasing then the following statements are equivalent: \begin{align} 1) & & & \sum_{n=0}^\infty (-1)^na_n \text{ ...
1
vote
1answer
34 views

Understanding how to justify inequality between limits

The problem goes as follows: Let $f_n$ be a sequence of functions with domain $[0,1]$ and let $B$ be a dense subset of $[0,1]$. Show that if $f_n$ is increasing, for every $n\in\mathbb{N}$ and ...
1
vote
1answer
35 views

Power series - interval of convergence

For $f(x) = \sum_{n=2}^{\infty} \frac{(x+1)^n}{n(n-1)}$ I have showed that $f'(x) = \sum_{n=1}^{\infty} \frac{(x+1)^n}{n}$ and that $f''(x)=\frac{-1}{x}$ at all points where f converges absolutely. ...
3
votes
1answer
37 views

Topologies conceptual confusion (topology of maximum norm/of pointwise convergence)

One of the questions from my lectures notes reads as follows: "Show that the identity map from $C[0,1]$ with the topology, $T_m$, induced by the maximum norm to the topology of pointwise convergence, ...
1
vote
1answer
27 views

Family of analytic functions from unit disk to the plane minus a line

Let $\mathcal F$ be the family of analytic functions on the unit disk $\,\mathbb D=\{z: \lvert z\rvert<1 \},$ such that $f[\mathbb D] \subset \mathbb C\setminus(-\infty, 0]$. Show that $\mathcal ...