Convergence of sequences and different modes of convergence.

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32 views

Divergence/Convergence of a series of positive numbers [closed]

How can I prove this series diverges/converges: $$ \sum_{n=2}^\infty \frac{1}{\ln(n)^{200}} \text{ ?} $$ I've already tried to use the integral, root and ratio tests, but it doesnt work...
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28 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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2answers
48 views

Show that $(s_n)$ is a Cauchy sequence

$$S_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... + \frac{(-1)^{n+1}}{2n-1} $$ Show that $(S_n)$ is a Cauchy sequence and hence that it converges to limit $L$. Show that $\frac{2}{3} < L ...
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2answers
61 views

Infinite Sequence Converging to x proof

I really don't understand how to do proofs on convergence at ALL. I know you're supposed to use $|xi - x|$ < $\epsilon$ but I have no idea how to apply this to this question: Show that if $x$ is ...
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2answers
70 views

If $\sum a_n$ converges, does $\sum a_n / 2^n$ converge as well?

If $\sum a_n$ converges, does $\sum \dfrac{a_n}{2^n}$ converge as well? I can't use differential or integral calculus for this.
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63 views

Does the series $\sum \sin(100n)$ converge? [duplicate]

Does the following series converge? $$\sum \sin(100n) = \sin(100) + \sin(200) + \dots$$
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2answers
35 views

Convergence of complex mercator series

I'm trying to find out for which $|z|=1$ the series $$\sum_{n=1}^\infty{}\frac{z^n}{n}$$ converges. It diverges for $z=1$ (harmonic series) and converges for $z=-1$ (alternating harmonic series). I ...
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1answer
27 views

Does Convergence of Maps Evaluated at Points Imply Convergence in Operator Norm?

Suppose that I have $T,T_n \in B_H$, for some Hilbert space $H$. Is the following implication true? $$ \|(T-T_n)x\| \rightarrow 0 \ \forall x\in H \ \Rightarrow \ \|T-T_n\| \rightarrow 0, \ \text{ie} ...
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1answer
30 views

Determining invariant probability measure and calculating $\lim_{n}p_{ij}^{(n)}$

Consider the Markov chain $(X_n)_{n\in\mathbb{N}_0}$ with state space $E=\left\{1,2,3\right\}$ and transition matrix $$ P=\frac{1}{2}\begin{pmatrix}0 & 1 & 1\\1 & 0 & 1\\1 ...
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34 views

counter-example: aboslute convergence => convergence in incomplete vector space

Is the following statement true? Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$ and $\sum_{k=0}^\infty \lVert x_k\rVert$ convergent. Then $\sum_{k=0}^\infty x_k$ is also ...
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1answer
183 views

Is this infinite series related to prime and composite numbers convergent?

I don't know whether this series converges: $$\frac{1}{4} - \frac{1}{5} + \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} - \frac{3}{11} + \frac{1}{12} - \frac{1}{13} + ...
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0answers
27 views

Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
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0answers
21 views

Taylor Polynomials: Estimating accuracy of an approximation f(x) ≈ Tn(x)

$f(x) = \sqrt{x},\space\space\space\space a = 4,\space\space\space\space n = 2,\space\space\space\space 4 \le x \le 4.7$ I approximated f by a Taylor polynomial with degree 2 at the number 4. ...
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1answer
66 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
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1answer
41 views

Where does this product of random variables converge to?

Consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ wich are independently normal distributed $N(0,\sigma^2)$. Set $M_0$=1 and $$ M_n =\exp \left( \sum_{i=1}^n X_i - ...
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2answers
51 views

Uniform convergence of $\sum f(x)^n$

Let $f:X\to\mathbb{R}$ be such that $\sup\{|f(x)|:x\in X\}<1.$ Show that $\sum_{n=1}^{\infty} f(x)^n$ converges and compute the sum.. Every value given by $f$ is less than one, then if ...
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5answers
163 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
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1answer
43 views

Convergence of the sequence $f_n(x)=\frac{1}{1+nx^2}$

I'm trying to find the convergence of $f_n$ and $f_n'$ where $f_n(x)=\frac{1}{1+nx^2}$. From the function if I derivate the result is $f'n= -\frac{2nx}{(1+nx^2)^2}$. To determine $f$ I have to take ...
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46 views

What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
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1answer
55 views

Show that $\lim_{n\to ∞} |a_n| = |a|$ if $a_n\to a$

Let $(a_n)$ be a convergent sequence with $\lim\limits_{n\to \infty} a_n = a$. Show that $$\lim_{n\to \infty} |a_n| = |a|$$ Then state and disprove the converse statement. In order to prove ...
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25 views

Define sequence and convergence

Define function f: $\mathbb{R}_+\rightarrow\mathbb{R} $ by: $ f(x)=\sqrt{\frac{x^2}{3}+\frac{18}{x}}$ 1) Show that $f'$ has one minimum/maximum, define $f'$s monotony conditions and sketch $f$. I ...
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244 views

Proving the sum of the reciprocals squared converges [duplicate]

I'm investigating the Basel Problem, and the sum to consider is: $\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}$ How can I show this converges? Using graphs/computer software is also fine, but ...
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1answer
43 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
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1answer
22 views

Convergence of the sequence of maxima of a function sequence

Suppose we have a compact set $K \subset \mathbb{R}$ and a sequence of continuous functions $f_n: K \rightarrow \mathbb{R}$. Let $f$ be the uniform (and hence continuous) limit of $(f_n)_n$. Assume ...
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1answer
33 views

Show that $f_n(x) = \log nx$ does not converge pointwisely

I am learning convergence of sequence of functions in $\mathbb{R}$ and I would like to show the following: Define $f_n : (0,1) \rightarrow \mathbb{R}, f_n(x)=\log nx$. Then $\{f_n(x)\}$does not ...
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52 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
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33 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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3answers
76 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does given integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we evaluate it's value?
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1answer
23 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
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1answer
59 views

Does the statement “$f = 0$ almost everywhere” depend on the measure that is defined?

I know the convention is to use the Lebesgue measure but is there ever a situation where we would interpret "$f(x) = 0$ almost everywhere" by using a different measure? For example, let $f(x) = 1$. ...
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34 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
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1answer
19 views

Convergence of the maxima of Cauchy random variables

Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of independent and identically distributed random variables such that $X_1$ has density function $f_{X_1}(x) = \frac{1}{\pi(1+x^2)}$, $x \in ...
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1answer
41 views

Product of a Convergent Series and Bounded Sequence

Let $a_n$ be a bounded sequence and $\sum_{n=1}^\infty b_n$ be a convergent series. Then $\sum_{n=1}^\infty b_na_n$ is convergent. I have found a counterexample to prove it false; If we let ...
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1answer
36 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
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2answers
35 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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0answers
47 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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25 views

Yet another interchange of limit and sum (revised)

A revised version of a similar question I asked before. $S_n$ are finite sets containing rational numbers in [0,1] for which $S_n\subset S_{n+1}$ and $\lim_n S_n = \mathbb{Q}\cap [0,1]$. ...
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41 views

Yet another interchange of limit and integral

$f_n$ are uniformly-bounded functions that converge pointwise on a dense subset of the compact set $D$ to a continuous function $f$. Is it true that $$\lim_{n\to\infty} \int_D f_n(x)\, \mbox{d}x = ...
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3answers
136 views

Show sequence is convergent and the limit

Given the sequence $$\left\{a_n \right\}_{n=1}^\infty $$ which is defined by $$a_1=1 \\ a_{n+1}=\sqrt{1+2a_n} \ \ \ \text{for} \ n\geq 1 $$ I have to show that the sequence is convergent and find ...
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2answers
56 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
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1answer
33 views

Prove that $\lim_{n\to\infty} f_n=0$ on $[0,\pi],$ where $f_n(x)=\cos^n(x)$

Prove that $\lim _{n\to\infty} f_n=0$ on $[0,\pi],$ where $f_n(x)=\cos^n(x)$ It's not difficult to see that when $x\in(0,\pi)$, we can take some large $N$ such that $\forall \epsilon>0$, $n\ge ...
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31 views

Convergence in Distribution

Suppose $X_n$ are $\mathbb{R}$-valued random variables. Then how would I be able to show that $X_n\rightarrow^D X$ where $X$ is also a $\mathbb{R}$-valued random variable iff $F_n(x)\rightarrow F(x)$ ...
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1answer
20 views

Alternating Series Test for Divergence

Alternating Series Test if the alternating series $$\sum^\infty_{n=1}(-1)^{n-1}b_n=b_1-b_2+b_3-b_4+b_5+...\;\;\;\;\;b_n\gt0$$ satisfies $$(\text{i)}\;\;b_{n+1}\leq b_n\;\;\;\;\;\text{for all ...
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2answers
70 views

Another way to find $\sum_{n=1}^{\infty}{\dfrac1{k^n}}$

Find sum $$S=\sum_{n=1}^{\infty}{\dfrac1{k^n}}$$ where $k\in\mathbb{Z^+}\setminus\{1\}$. In my book they used limits to find this sum. They wrote $$S=-1+\sum_{n=0}^{\infty}{\dfrac1{k^n}}$$ and then ...
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1answer
38 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
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1answer
51 views

Series convergence and limit

Can this question please be deleted by some admin? Thanks.
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1answer
26 views

Finding a function greater or less than factorial function

suppose we are given the sequence: $a_n = (-1)^n\frac{1}{n!}$ using squeeze theorem find the limit: $$\lim_{n\to \infty} (-1)^n\frac{1}{n!}$$ using the squeeze theorem. For factorials, $a_n$ how ...
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85 views

Determine if $\displaystyle \int_3^{\infty}\frac{x+1}{\sqrt{x^4-x}}\,dx $ converges

Determine whether the following integral is convergent or divergent without evaluating it. (Whichever answer is correct, you must show why it is true.) $$ ...
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4answers
110 views

Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?

$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$ The first one is an alternating series, so it would just be: $$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow ...
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1answer
29 views

convergence of an alternating series without $a_n$

I have an infinite series that goes like: 3-$\frac{69}{5}$+$\frac{834}{25}$-$\frac{7734}{125}$+$\frac{62109}{625}$-$\frac{455859}{3125}$+$\ldots$ I can generate more terms of this series if needed. ...