Convergence of sequences and different modes of convergence.

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Root test for convergence: $\displaystyle{\lim_{n\to\infty} (a+bi)^n}$

$$\lim_{n\to\infty} (a+bi)^n$$ where $i$ is the imaginary unit. I'm having trouble with this question. I get to $a+bi$ but I have no clue how to finish it in order to determine if it converges ...
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1answer
28 views

To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $\lim_{x \to 0} \frac {2^x-1} x$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that if ...
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1answer
38 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
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0answers
20 views

Visual understanding of radius of convergence for complex power series

I am examining Needham's proof (Visual Complex Analysis 2.III.2) that $\sum c_k z^k$ converges at $a\ne 0$ implies convergence, in fact absolute convergence, for $|z| < |a|.$ Despite the book's ...
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4answers
72 views

Convergence or divergence of the series $\sum\limits_{n = 1}^{\infty} \sin(\pi/n)$

Let $ u_{n} = \sin \! \left( \dfrac{\pi}{n} \right) $, where $ n \in \Bbb{N} $, and consider the series $ \displaystyle \sum_{n = 1}^{\infty} u_{n} $. Which of the following is/are true? (a) $ ...
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1answer
19 views

Does weak-$\ast$ convergence with an exponential rate imply convergence of measures of sets with the same rate?

Assume that $\mu_n \to \mu$ in the weak-$\ast$ topology with the following rate for any compactly supported continuous function $f$: $$|\mu_n(f) - \mu(f)| \leq C_f e^{-n}.$$ Can we replace $f$ with ...
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2answers
27 views

Convergence Events with States

Ignatz repeatedly rolls a fair $6$-sided die. What is the probability that he rolls his first $5$ before he rolls his second (not necessarily distinct) even number? I don't know what to do about the ...
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2answers
137 views

The Typewriter Sequence

The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e. Could someone explain why it does not converge to zero a.e.? Note: the ...
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3answers
56 views

Given $\phi \in C^{1,b}(R)$, find $\phi_n$ countably piecewise affine functions whose derivatives converge to $\phi'$ uniformly where differentiable

Let $\phi \in C^{1}(\mathbb R)$ with bounded derivative. I am trying to build $\phi_n$ a sequence of countably piecewise affine functions, s.t. $\phi_n'$ converges uniformly to $\phi'$ on $N^c$, where ...
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1answer
19 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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1answer
53 views

How is this trival?

I was reading an article today and on section 2 it is indicated that if we are given a Radon Measure $\mu$, and a real $p$ then fast convergence entails trivially almost sure convergence, where fast ...
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1answer
24 views

Example of a pointwise convergent functional sequence that is not compactly convergent.

I'm looking for an example of a pointwisely convergent functional sequence $\{f_n\}_{n \geq 0}$ (where $f_n:\mathbb{R}\to\mathbb{R}$) that is not compactly convergent. I'm not sure if it is even ...
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3answers
26 views

Is this a counter example for a comparison test for sequences?

I’ve recently started learning about sequences and convergence and divergence, and I came across the comparison test for sequences. What I have is that: What if $a_n$ is defined as a periodic ...
2
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1answer
45 views

Convergence of an integration $t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy$

When I am reading Brian Hall's "Quantum Theory for Mathematicians", I came across an integration (frequently appeared in physics textbooks) $$t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy.$$ The ...
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0answers
37 views

Coconvergent topology basis?

Consider the space $X=\{\frac1n:n\in\mathbb N_+\}$, with the "coconvergent topology": $$\mathcal O=\{A:(A=\varnothing)\lor(\sum_{x\notin A}x<\infty)\}$$ That is, a nonempty set is open iff its ...
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1answer
49 views

Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$

I have trouble with the following sum: $$S=\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$$ Since $a_n=\frac{2}{n}$ decreases monotonically and tends to $0$, it converges by Liebniz criterion. Then, by ...
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21 views

Convergence of a sequence using Banach contraction principle: $x_{n+1}=\frac{1}{x_n+a},x_1>0,a \in \mathbb{R},n=1,2,…$

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
3
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2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
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3answers
32 views

Prove that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$

Could someone please show me the proof that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$ I have no idea where to begin with this one. Thanks.
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3answers
34 views

Show that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$

Could someone please show me the algebraic steps in showing that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$? As the way I see it $\frac{1}{2}x^{\frac{2}{x}} ...
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0answers
34 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
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18 views

pick a sequence of parameters to preserve original convergence

Suppose for any fixed $\delta >0$, we have $$X_n(\delta) \overset{\mathbb{P}}\to 0 \quad \text{as}\ \ n \to \infty.$$ Does there exist a sequence $\delta_n \to 0$ such that $$X_n(\delta_n) ...
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2answers
67 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
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1answer
40 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
3
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0answers
24 views

Limit of $\frac{1}{2^n}\sum_{n=0}^{2^n-1}f(e^\frac{2k\pi}{2^n}i)$ for a complex-analytic function

Let consider a Laurent series $\displaystyle{ \sum_{k\in\mathbb Z}a_kz^k }$ with complex coefficients and converging inside the annulus $A=\{\ z\in\mathbb C\ |\ r<|z|<R\ \}$, with ...
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1answer
39 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
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1answer
39 views

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(n)}$ converge?

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(p)}$ converge? I'm trying to use the ratio test but I can't get a simple term in which use limit easily enough.
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1answer
16 views

existence of a sequence of continuous functions with two conditions

$\displaystyle \int_0^1 \lim_{n\to\infty} f_n(x)\,dx = \lim_{n\to\infty}\int_0^1f_n(x)\,dx $ There is no function $\,g:\left[0,1\right]\to \mathbb R\,$ lebesgue integrable such that $\,\left\lvert ...
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33 views

Show that $S_n/n$ converges almost surely if $S_{2^n}/2^n$ converges almost surely

Let $X_n$ be independent random variables such that $\dfrac{S_n}{n}\to0$ in probability and $\dfrac{S_{2^n}}{2^n}\to0$ almost surely. Show that, $\dfrac{S_n}{n}\to0$ almost surely. Here ...
2
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0answers
38 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
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36 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
2
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1answer
43 views

Convergence of random variables taking integer values

Let $X_n$ be random variables taking integer values, and let $X_n\to X$ in distribution. Show $X$ also takes only integer values. $P(X_n=j)\to P(X=j)$ for each integer $j$. $\displaystyle ...
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1answer
18 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let ...
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2answers
73 views

When $\lim_{n\to\infty} \mathbb P(A_n) = \mathbb P(\lim_{n\to\infty} A_n)$? [closed]

When is the following statement true for a sequence $(A_n)_{n\in\mathbb N}$ of events: $$\lim_{n \to \infty} \mathbb{P} (A_n) = \mathbb{P} \left( \lim_{n \to \infty} A_n \right)?$$
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0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
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0answers
108 views

Is $1+2+3+4+\cdots=-\frac 1{12}$ true? [duplicate]

Hello (it's my first post here!), I have a strange question. I heard that (under certain conditions): $$ 1+2+3+4+\ldots=\sum_{k=1}^{\infty}k=-\frac{1}{12} $$ Is it REALLY true? And - if yes - how to ...
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53 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
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1answer
53 views

Proof of Product Rule for Sequences using definition of infinitesimal and properties of infinitesimal sequences.

I have been trying to understand this proof for the product rule of sequences, where the author makes use of some properties for infinitesimals, to prove this theorem. This is quite a long question, ...
0
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1answer
76 views

Almost sure convergence of nonrandom sample

This is a question about almost sure convergence. Consider the following set-up: There are $B$ banks. Each has size $S_{b}$, which follows a size distribution $f_{S}$ with mean E[S]. $f_{S}$ is ...
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3answers
63 views

Convergence and Limit of a Sequence

I am looking at the sequence and I'm trying to see what happens as $n\to\infty$ $$ a_n = \frac{\prod_{1}^{n}(2n-1)}{(2n)^n} = \frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{(2n)^n} $$ By inspection, I can ...
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4answers
69 views

Uniqueness of a Limit epsilon divided by 2?

I have been reading about this theorem in a book called 'Calculus: Basic Concepts for High-schools', it is a very good book (so far) and I can highly recommend it. Well the author goes on to prove ...
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3answers
37 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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38 views

Convergence and limit of a sequence $x_n=1+\frac{x^2_{n-1}}{2},n\ge2,x_1=\frac{3}{8}$

$x_{n+1}-x_n=\frac{x^2_n-2x_n+2}{2}>0$ sequence is increasing. I don't know how to prove that it is bounded. Limit should be $\frac{1}{2}$
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1answer
29 views

A question on convergent series of positive terms

Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
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1answer
21 views

Completeness of a system: For all $n$ and within an interval?

Why is the system $\sin((2n-1)x)$ for $n=1,2,\cdots$ complete in $L^2[0,\frac\pi2]$? This means that the Euclidean norm converges for $n=1,2,\cdots$ and for all $x\in[0,\frac\pi2]$ How does one prove ...
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
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0answers
51 views

How to prove that this series converges. [duplicate]

I want to prove that this series converges: \begin{equation} 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... \end{equation} I normally use this one as a standard series to test other series for their ...
0
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1answer
31 views

Example of each $\{f_n\}$ Riemann integrable such that $\sum f_n$ converges point-wise to $f$ which is not Riemann-integrable

I am looking for an example of a sequence $\{f_n\}$ of real valued Riemann integrable functions on a closed bounded interval such that $\sum f_n$ converges point-wise to a function $f$ which is not ...
1
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5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
4
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5answers
102 views

Prove that $\sum\frac{n+1}{(n+2)n!}$ converges

Show that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{n+1}{(n+2)n!}$ converges, using the integral test. I noticed that $\displaystyle\sum\frac{n+1}{(n+2)n!} = \sum\frac{(n+1)^2}{(n+2)!}$, but ...