Convergence of sequences and different modes of convergence.

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Limit of sequence partially applied to a function

Let $(a_n)_{n\in\mathbb N}$, $(b_n)_{n\in\mathbb N}$, $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ be real-valued sequences and $f:\mathbb R\to \mathbb R$ monotone with ...
2
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1answer
12 views

Investigating uniform convergence of a sequence

I am trying to determine if the sequence $f_n$:= $\frac{x^{2n}}{1+x^{2n}}$ is uniformly convergent on $D_1:=[-q,q],0<q<1$, and $D_2:= (-\infty,r] \cup [r,\infty),r>1$. I have determined that ...
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0answers
25 views

Bounded derivative and Taylor's polynomial

Given function $f:\mathbb{R}\rightarrow \mathbb{R}$ of class $C^{\infty}$ such as $\forall_{n\ge2015}{\forall_{x\in\mathbb{R}}{|f^{(n)}(x)|\le7}}$ a)prove that sequence of functions $\{T_{n,f,0}\}$ ...
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1answer
19 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
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2answers
62 views

Proving that if $p_{n}$ converges then $|p_{n}| $ converges

EDIT1: Prove using the definition of a converging sequence in a metric space, that the convergence of the sequence $\left \{ p_{n} \right \}_{n=1}^{\infty}$ implies the convergence of the sequence ...
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1answer
44 views

A sequence that converges weakly but not in the Cesàro sense

Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$, and let $\{x_n\}_{n=1}^\infty\subseteq H$, $x\in H$. I'm using the following definitions: ...
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0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
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1answer
34 views

When is it ok to use a sequential limit in place of a continuous limit?

I am working through some Lebesgue integral problems, and I've come across a few instances where I would like to use the dominated/monotone convergence theorems, but the limit is continuous, and I'm ...
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1answer
31 views

Divergent or convergent but how ??

I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...
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4answers
55 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then ...
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1answer
49 views

Is the extended real line a metric space?

I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded. My question is: Is ...
4
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2answers
121 views

Paradox or error in design?

Currently I'm writing a homework for my school. I've made an experiment built this way: There is a laser pointed at a half reflecting mirror which reflects 50% at a wall. The other half cross the ...
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2answers
76 views

Why is convergence required for a series to be differentiable? [on hold]

Since moderators marked this question as "unclear" I will repeat the title maybe this won't be marked. Why is convergence required for series to be differentiable? I want intuitive explanation - not ...
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0answers
16 views

Average of Convergent Sequences Proof [duplicate]

Show that if ($x_n$) is a convergent sequence, then the sequence given by the averages: $y_n$=($x_1$+$x_2$+...+$x_n$)/n also converges to the same limit. I know that for all $\epsilon$$>$0, ...
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3answers
49 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
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1answer
35 views

Floor and Ceiling Series (I) [on hold]

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
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1answer
36 views

Converging or diverging?

I did try a lot of approaches to this question but I am yet to get a correct one that is if $$A_n = 1/((\log{n})^{\log{n}})$$ then is the summation $\sum A_n$ convergent or divergent ?
3
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2answers
47 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
0
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1answer
21 views

Proving convergence of a sequence through a polynomial

Let ($x_n$)$\rightarrow$x and let $p(x)$ be a polynomial. (a) Show $p(x_n)$$\rightarrow$$p(x)$. (b) Find an example of a function $f(x)$ and a convergent sequence ($x_n$)$\rightarrow$x where the ...
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1answer
24 views

Convergence Proof Help?

Question: Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence $(x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ ...
4
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1answer
85 views

Find a convergent solution for $a$

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you ...
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2answers
35 views

Convergent Summation Proof

I'll preface and apologize for the somewhat lengthy intro. Alright, so recently I've been self-studying summation and found interest in convergent series. I was looking at some fairly common series ...
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0answers
34 views

Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$ where the $f_n$ are measurable functions? ...
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2answers
46 views

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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2answers
56 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 ...
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1answer
41 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
30 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 $a$ implies absolute convergence (and hence convergence) for $|z| < |a|.$ Despite the book's ...
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4answers
77 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) $ ...
1
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1answer
21 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 ...
0
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2answers
33 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 ...
4
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2answers
141 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
57 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
20 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
26 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
28 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
48 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
39 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
50 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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0answers
21 views

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

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
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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
33 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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0answers
20 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
69 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
41 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
41 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? ...
0
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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.