Convergence of sequences and different modes of convergence.

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Sequences of random variables converging in probability to the same limit a.s.

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, ...
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15 views

Uniform convergence on compact space

Let there be two functional squences $$a_n(x)=\sqrt[n]{x} \quad \textrm{ for $x\in(0,\infty)$}$$ $$b_n(x)=\sum_{k=0}^{n}x^k(1-x)^k=\frac{1-x^{n+1}(1-x)^{n+1}}{x^2-x+1} \quad \textrm{ for $x\in ...
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44 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [on hold]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
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1answer
22 views

Uniform and pointwise convergence

So we had the pointwise and uniform convergence, and I do get that a sequence of function can converge to a function, just like ordinary sequences do. But what I don't quite get is this pointwise and ...
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1answer
20 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
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24 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
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1answer
15 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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2answers
647 views

Prove $\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty$ for an increasing sequence $a_n$ of positive integers [duplicate]

The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that: $$ \sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty $$ While I have ...
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Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
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1answer
572 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
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1answer
39 views

Prove point-wise convergence for the sequence $\{f_k\}$ of functions

Consider the sequence $\{ f_k\}$ of functions in $C[0,1]$ defined by $$ f_k(x) = \begin{cases} 0, & 0 \leq x \leq \frac{1}{k} \\ 2(k^{3/2} x - k^{1/2}), & \frac{1}{k} \leq x \leq ...
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3answers
78 views

Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge?

I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I ...
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1answer
31 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
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A different notion of convergence for this sequence?

I was thinking about sequences, and my mind came to one defined like this: -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ... Where the first term is -1, and after the nth occurrence of -1 in the ...
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2answers
46 views

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? [duplicate]

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? This is what I have done, since $0\le x\le 1$ $ x=0$$$f_n(0)=0-0^n=0$$ $x\to \infty$ ...
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1answer
19 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) ...
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29 views

Convergence of $\sum \frac{a_n}{1+a_n}$ implies convergence of $\sum a_n$ for positive $a_n$. [duplicate]

I need to prove or disprove the statement. I think the statement is true. My attempt at a proof: From the definition of convergence: $$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad ...
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0answers
14 views

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$ converges if the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \rvert$ converges too and the ...
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1answer
21 views

Should monotone convergence theorem say uniformly bounded?

saz pointed out to me the difference between bounded and uniformly bounded: $Y$ is uniformly bounded: there exists $C>0$ such that $|Y_n| \leq C$ for all $n \in \mathbb{N}$, i.e. $$|Y_n| ...
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0answers
7 views

sum converge, matrix, norm

Let $A_j$ be a sequence in $\mathbb{C}^{n\times n}$. Show that $ \sum_{j=0}^\infty A_j$ converges if $ \sum_{j=0}^\infty ||A_j||$ does.($||A||= sup_{|x|=1} |Ax|$ with euclid norm) Hello, Be ...
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2answers
29 views

Sum of reciprocals of prime-index-primes

Let $p_1=2$, $p_2=3$, $p_3=5$, $\ldots$ be an enumeration the prime numbers. If $q$ is a prime number, we call $p_q$ a prime-index-prime. A list of prime-index-primes can be found here. My question ...
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2answers
146 views

Example 3.53 in Baby Rudin

Here's Example 3.53 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. Consider the convergent series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ...
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2answers
47 views

Convergent subsequence in a bounded sequence of a complete metric space

Consider a complete metric space E with the following property: If $x_n$ is a bounded sequence, then $\forall \epsilon > 0$, $\exists i,j , i \neq j$ such that $d(x_i,x_j) < \epsilon$. ...
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2answers
58 views

Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
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1answer
33 views

If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

Let $a_{n} \geq 0$ and $b_{n}>0$ for each $n$ in $\mathbb{N}$ and suppose that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} ...
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23 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
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33 views

Convergence from another series

Suppose that both $$\sum_{j=0}^n a_j^2$$ and $$\sum_{j=0}^n b_j^2$$ are convergent. Show that $$\sum_{j=0}^n a_jb_j$$ converges absolutely. Ok, so my final exam is tomorrow and I have been working on ...
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Where does the following series converge? [on hold]

Using integrals or by any other method find: $\lim_{n \rightarrow\infty} \sum_{i=1}^{n}\frac{1}{n+i}$
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Uniformly convergent Laurent Series [on hold]

Why does a Laurent Series with positive and negative parts converge uniformly only on compact sets?
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42 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
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1answer
24 views

Convergence of $f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1]$

$(f_n)$ is a succession of functions $$ f_n : [-1,1] \rightarrow \mathbb{R} \\ \\ f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1] $$ Punctual convergence $\forall x \in ...
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Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
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12 views

Limit laws for power of convergent sequence

Let $ \{ a_n \}_{n=m}^{\infty}$ be convergent sequence of real numbers such that $\lim _{n \rightarrow \infty}a_{n}=x$. Can we say that $\lim _{n \rightarrow \infty}a_{n}^q=(\lim _{n \rightarrow ...
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20 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
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For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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Sequence converging to a closed set

How can I prove that A sequence generated by a particular algorithm converges to a set? i.e irrespective of the starting point the sequence converges to any of the points in a set, or oscillates ...
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Determine whether the following sequences (fn) converge uniformly, pointwise, or neither:

Determine whether the following sequences $(f_n) \in F(E, \mathbb{R})$ - where E is a set - converge uniformly, pointwise, or neither: a) $f_n(x) = \frac{n^2x} { 1 + n^2x^2}$ on set $E = \mathbb{R}$ ...
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How to prove that $\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? [on hold]

Could you tell me how to show if $p>0$ then$\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? (+clues) 1.put $\sqrt[n]{p}=1+h_{n}$ 2.Bernoulli's inequality If you don't mind, use the clues to prove it.
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572 views

Confused about series and testing for convergence/divergence?

I'm finding it quite difficult to understand the idea of series and limits to test for divergence or convergence. Perhaps more so in finding such a limit. I have the series $$\sum_{n=1}^\infty ...
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25 views

A non unconditionally convergent series? [on hold]

A series $\sum_{n\ge 1}a_n$ in a Banach Space $X$ is said to be unconditionally convergent if it converges for any rearrangement of its terms. Denote by $(e_n)$ the canonical basis of $\ell_2$, ...
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45 views

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$ I applied nth term test and was inconclusive. I tried ratio test but I don't know how to evaluate the limit. I think it is 1 therefore ...
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1answer
40 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
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3answers
72 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
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1answer
30 views

Find convergence domain of the integral

Find convergence domain of $$\int_0^\infty \! \frac{\cos^2{x}}{x^p} \, \mathrm{d}x$$ I've tried to use $\frac{\cos^2{x}}{x^p} < \frac{1}{x^p}$, but $\int_0^\infty \! \frac{1}{x^p} \, \mathrm{d}x$ ...
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When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
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1answer
38 views

Investigate convergence of $\sum_{n=2}^\infty \frac{1}{n(n-1)}$

Investigate convergence of: $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ Now I know by $p$-series test this summation converges however, is there a way to prove that this series converges by some ...
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0answers
57 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
3
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2answers
46 views

Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
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1answer
29 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
6
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1answer
39 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...