Convergence of sequences and different modes of convergence.

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Convergence of $\frac{1}{4n^2-1}$ on the range of n=1 to infinity.

I know that the series converges to $\frac{1}{2}$. Can someone please show the steps to prove this though and what convergence test can be used? Thanks.
8
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2answers
234 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
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1answer
521 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
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2answers
222 views

Finding radius of convergence for series

Find the radius of convergence of the given power series: $$\sum _{n=1}^{ \infty} \frac{(-1)^n x^{5n+2}}{5n+2}$$ Through the ratio test, I can get it down to $$ \frac{(-1)^n+1 ...
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1answer
11 views

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 ...
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2answers
125 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
55 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
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1answer
697 views

Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
8
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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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4answers
132 views

Which (convergent) series can one find the sum of?

I know about geometric series and how one can find the sum when they are convergent. I also have heard that one can prove that the $p$-series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ has sum $\pi^2/6$ ...
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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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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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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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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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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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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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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 ...
3
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1answer
173 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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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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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 ...
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1answer
45 views

Good problem book for convergence concepts in probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
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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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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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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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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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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 $\{ f_{n} \}$ ...
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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 ?
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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
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 $(x_n)$ ...
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2answers
164 views

Unnecessary condition of Lebesgue's monotone convergence theorem?

Rudin RCA p.21 Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n\}$ be a sequence of measurable functions on $X$ and suppose that (a) $0\leq f_1\leq f_2\leq\cdots\leq\infty$ for every $x\in X,$ ...
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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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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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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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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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1answer
55 views

Series using comparison test

The series is as shown $$\sum\limits_{n=1}^\infty\dfrac{\tan(1/n)}{n^{0.5}}$$ Using P-series, 0.5<1 therefore it is divergent $$\frac{1}{n^{0.5}}$$ It appears that i can't use P-series What test ...
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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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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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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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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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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) $ ...
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157 views

Does the harmonic series converge if you throw out the terms containing a $9$?

I found this very amusing comic on the internet the other day: The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this ...
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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 ...
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4answers
72 views

How to find the limit of a Sequence $ f(n) = \frac{16f(n-1)-2}{32} $

I have this sequence: \begin{equation} f(n) = \frac{16f(n-1)-2}{32} \end{equation} with \begin{equation} f(0) = 1 \end{equation} I want to find it's limit while n goes to infinity and therefore check ...
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94 views

A question about convergence of $b_n=f\left(\frac 1{n^2}\right)$

Suppose $f(x)$ is some function with domain [0,1] and $\sum_{n\ge1}f\left(\frac 1n\right)$ converges, than $a_n=f\left(\frac 1n\right)\to 0$ but does $b_n=f\left(\frac 1{n^2}\right)$ also converge to ...
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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 ...
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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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1answer
53 views

Convergence in $L^{p_1}$ and $L^{p_2}$

Suppose $f_k$ is a sequence of $\mu$-measurable function. Let $p_1$ and $p_2\in[1,\infty)$, and $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose that there exists $g\in L^{p_1}$ and $h\in L^{p_2}$ such that ...
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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 ...