Convergence of sequences and different modes of convergence.

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(Resolved) Does the sum of a subset of the Harmonic sequence converge iff its density approaches 0?

Update: This question has been resolved. I have made some mistakes in this post. I will leave my post here for readers to find out my mistakes. I have noticed that the post is a bit too long. So if ...
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10 views

Can you describe two graphs as convergent over an interval?

When describing two graphs that seem to follow the same path (literally, not parallel as there is an intersect) is it mathematically correct to say that they seem to converge over an interval? Or to ...
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1answer
48 views

Pointwise and uniform convergence of a series of functions

Define a sequence of functions on $[0,\infty)$ such that $\forall n\in\mathbb{N}$, $$ f_n(x)\triangleq \begin{cases} 1 & x\in[n,n+\frac{1}{n}]\\ 0 & \text{otherwise} \end{cases} $$ Does the ...
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1answer
25 views

Using Comparison Test for ∑(1/(n^2-ln(n)))

Good evening all! I have been stuck on this problem for way too long, and figured I could use a little push in the right direction. I have this series: $$\sum \frac{1}{n^2 - ln(n)} $$ and would ...
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1answer
44 views

Radius of convergence of $\sum\limits_{n}c_{n}z^{n^{2}}$ given that the radius of convergence of $\sum\limits_{n}c_{n}z^{n}$ is finite and nonzero

I know that the radius of convergence of a given power series $\sum_{n=1}^{\infty}c_{n}z^{n}$ is $R$, where $0<R<\infty$. Given this information, I need to find the radius of convergence of ...
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1answer
21 views

Convergence of series $a^x$ where $a < 1$

I'm checking for convergence of the series $$\sum_{x=0}^{\infty}a^{x}$$ when $a<1$ and $a>0$. My analysis, $a_0 = 1, a_1 = a<1, a_2 = a^2 < 1\dots a_{\infty} = ...
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53 views

Sequence of functions: Convergence

For each of the following, give an example of a sequence of functions $f_n$ that converges to f A. uniformly but not in the mean square sense. B. in the mean square sense but pointwise nowhere. I ...
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1answer
51 views

Find the value of sum (n/2^n)

I have the series $\sum_{n=0}^\infty \frac{n}{2^n}$. I must show that it converges to 2. I was given a hint to take the derivative of $\sum_{n=0}^\infty x^n$ and multiply by $x$ , which gives ...
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12 views

Determining the Radius of Convergence of a power series

Consider the series $\sum_{n=0}^\infty a_n (x-c)^n $. Let $\lambda = lim_{n\rightarrow\infty} \frac{a_{n+1}}{a_n}$. Show that if $\lambda$ exists in $[0,\infty]$, then $R = \frac{1}{\lambda}$, where ...
2
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1answer
36 views

Rate of convergence of Cesàro means

For a sequence $a_n = O(n^{-1/2})$ as $n\to\infty$, consider the corresponding Cesàro means $b_n = \frac{1}{n} \sum_{j=1}^n a_j$. Is it possible to derive the rate of convergence for the sequence ...
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47 views

Prove the convergence and find the sum of series $\sum\limits_{n=1}^{\infty}\left(n^3\sin\frac{\pi}{3^n}\right)$.

We know that $0<\sin\frac{\pi}{3^n}\le\frac{\sqrt 3}{2},\forall n\ge 1$. How to find the boundary for $n^3\sin\frac{\pi}{3^n}$ (how to use comparison test here)? I tried using the ratio test, but ...
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1answer
36 views

Show that the closure of $C_c(X)$ is $C_0(X)$.

Let $(X,T)$ be a topological Hausdorff space. By $C_b(X)$ denote the continuous bounded function $f\colon X\to\mathbb{R}$, by $C_c(X)$ the continuous functions $f\colon X\to\mathbb{R}$ which have ...
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12 views

Absolute-Normal convergence of a sum

I have a sum which is normal convergent and I know the limit-function of the sum, does it implies that the sum of the absolute values of each term converges to the absolute value of the function?
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1answer
22 views

Annular regions for which this Laurent series converges

Given the Laurent series $$\sum_{n= - \infty}^{\infty} \frac{z^n}{3^n + 1}$$ Find the annular region for which it converges. I'm struggling to find any similar examples or where to begin for this.
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1answer
28 views

Prove that the sequence is convergent and find its limit.

Prove the sequence:$$y(n) = (y(n-1) + 2y(n-2))/3 \text{ for } n > 2 \text{ and }y(1)<y(2)$$ is convergent and find it's limit. My progress so far So far, I have been able to prove that that ...
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14 views

What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits

I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ...
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1answer
23 views

An examination of rates of convergence of the series for $\pi$

I got this topic for project "An examination of rates of convergence of the series for $\pi$". My question is which relative formulas or math knowledge I should research?
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13 views

Convergence of Probability Measures and the Dirac Delta

Suppose that we define the measures on $\mathbb{R}$: $\displaystyle\mu_{n}=\sum_{j=0}^{n}\frac{1}{2}(\delta_{a_{j}}+\delta_{-a_{j}})$ For $\delta$ being the Dirac delta function, and for $a_{j} \in ...
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16 views

Prove improper integral converges

I'm studying the behaviour of the Bessel function as $x \rightarrow \infty$ part of the assignment requires me to prove the following: Prove the improper integrals $$\int_x^\infty ...
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1answer
49 views

For what value of $x$ the following series will converge $\sin x+2 \sin \frac x 3+4\sin \frac x 9+\ldots $?

For what value of $x$ the following series will converge $\sin x+2 \sin \frac x 3+4\sin \frac x 9+8\sin \frac x {27}+\ldots $? Work: \begin{align} \sin x+2 \sin \frac x 3+4\sin \frac x 9+8\sin \frac ...
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2answers
49 views

Convergence is uniform on every compact subset of $\Omega$?

Suppose $\{f_n\}$ is a uniformly bounded sequence of holomorphic functions in $\Omega$ such that $\{f_n(z)\}$ converges for every $z \in \Omega$. Does it necessarily follow that the convergence is ...
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27 views

Having trouble showing existence of pointwise convergent subsequence of a sequence of real-valued pointwise bounded functions on a countable set.

I am working on the following exercise from Royden's Real Analysis (Chapter 10, Section 10.1 on the Arzela-Ascoli Theoerem): Let $S$ be a countable set, and $\{ f_{n} \}$ a sequence of real-valued ...
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22 views

Applying Minimization algorithm on Rosenbrock function!

Why does rosenbrock function not converge using gradient method? But converges using Newton's method?
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23 views

Does a convergent sequence of norms of a vector space always converge to a norm?

I thought that just as the sequence of norms $||x||_p :\mathbb{R}^n \mapsto \mathbb{R}$ converges to $||x||_{\infty}$ maybe there is some result that proves that every convergent sequence of norms of ...
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90 views

How to find power series with the following interval of convergence?

$a) [-1,1] $ (conditionally convergent both at $-1$ and $1$) $b) [e,\ \pi) $ $c)$ center at $x=-\sqrt{2}$ and interval of convergence $(-\infty, \infty)$ I think to solve the question, we basically ...
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32 views

Convergence of random variable divided by constant that goes to inifinity

I have read in some book the following Lemma: For any random variable $X$ and for any sequence $c_n$, with $\displaystyle\lim_{n\to\infty} c_n = \infty$ the following is true: ...
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1answer
52 views

Convergence of injective functions

On the complex plane, it is true that if $\{ f_n\}$ is a set of holomorphic injective functions on the complex plane defined on a connected open set $\Omega$, which are convergent to $f$ uniformly on ...
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1answer
54 views

convergence of sequence in a Distribution

Let $\varphi \in D$ be a test function on $\Bbb{R}$. Is the sequence $f_n(x)=\frac{1}{n}\varphi(\frac{x}{n})$ convergent in the test function space $D$? What is the limit? please a hint to start.
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43 views

Expanding Laurent series

Expand the following Laurent series centered at the origin. Indicate the annulus of convergence. $$\frac{1}{(z^2+1)(z-2)}$$ I am attempting to decompose using partial fractions as I did on previous ...
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1answer
26 views

Liminf of sequence of sets $(A_n\cap B_n)$

Using the definition: $$\liminf_{n \to \infty} A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k$$ I did some set algebra (wich I'm not sure is right): $$\liminf_{n \to \infty} (A_n\cap B_n) ...
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1answer
42 views

Does this complex power series converges on the unit circle?

Prove that the series $$ \sum_{n=1}^\infty \frac{(-1)^{[\sqrt n]}}{n}z^n$$ converges on $\partial B(0,1)$.Where $[x]$ implies the greatest integer that is not bigger than $x$. It is easy to prove ...
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32 views

relationship between convergence of a sequence and its corresponding series.

I'm preparing for my calculus exam and I'm unsure how to approach these type of questions . If the sequence $a_n$ is convergent/divergent what can we about the corresponding series $\sum_{n}a_n$? Is ...
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40 views

Minimal Requirements for Convergence of Series

It is a standard fact that the series $$\sum_{n=1}^{\infty} \frac{1}{a_n}$$ converges if $a_n \gg n^{1+\epsilon}$ for some $\epsilon>0$. Moreover, it will diverge if $a_n \ll n\log(n)^A$ for some ...
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50 views

Does this sequence of polynomials converge to the square root function?

Taken from Lang's R & F Analysis (p.60). For some reason I can't see why, for $t \in [0,1]$, the following is true for all natural numbers $n$ (by an inductive argument): $$0 \leq \sqrt{t} - ...
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46 views

Using the Root Test to Solve $\sum_{k=1}^\infty(\frac{k^{2}}{2^{k}})$

$\sum_{k=1}^\infty(\frac{k^{2}}{2^{k}})$ So I've taken the $kth$ root of the numerator and denominator and got: $\sum_{k=1}^\infty(\frac{k^{\frac{2}{k}}}{2})$ $k^{\frac{2}{k}}$ as k approaches ...
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5answers
81 views

Does the sequence $\frac{n!}{1\cdot 3\cdot 5\cdot … \cdot (2n-1)}$ converge?

I'm trying to determine if this sequence converges as part of answering whether it's monotonic: $$ \left\{\frac{n!}{1\cdot 3\cdot 5\cdot ... \cdot (2n-1)}\right\} $$ First, I tried expanding it a ...
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26 views

Find the interval for the power series: $\sum_{n=0}^{\infty}\frac{(n+2)!}{n!}\left(x+2\right)^n$

$$\sum_{n=0}^{\infty}\frac{(n+2)!}{n!}\left(x+2\right)^n$$ To solve I used the ratio test and just wanted to see if my method was correct: $$\lim\limits_{n \to ...
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19 views

Analytic method for determining if a function will be negative.

I am learning the alternating series test and one of the conditions is that the function must be decreasing. One way to figure this out is to take the derivative and asses whether it is less than 0. ...
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1answer
37 views

Prove that $\{f_n\}$ converges to some $f$ in mean.

Let $f_m:[0,1]\mapsto\mathbb{R}$ be defined as for $m\in\mathbb{N}$ such that $\exists m = 2^n + k,\ n\in\mathbb{N},\ k\in\{0,1,2,\dots,2^n-1\}$ $$ f_{2^n+k}(x) = \begin{cases} 1 & ...
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Bounds for infinite series involving exponentials

Let $$ S(a,b):=\sum_{j=1}^\infty \exp(-a j^b ) , \quad a,b > 0 $$ which (due to monotonicity) can be bounded by $$ S(a,b) \leq \int_0^\infty \exp(-a x^b ) \, \mathrm{d} x = ...
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1answer
20 views

Interchange of limits and uniform limits

Analysis Vol II by Terence Tao I have no trouble proving the uniform limit theorem using the given hint $d_Y(f(x),f(x_0)) \le d_Y(f(x),f^{(n)}(x)) + d_Y(f^{(n)}(x),f^{(n)}(x_0)) + ...
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1answer
25 views

$L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
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1answer
38 views

Stochastically converging Bernoulli-related sequence

$X_1, X_2, .. ,X_n$ are independent Bernoulli random variables $X_i$ ~ BIN(1,$p_i$). Let $Y_n$ = $\sum_{i=0}^n(X_i-p_i)/n$. The problem is to show that the sequence $Y_1, Y_2, \ldots , Y_n$ ...
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16 views

Rate of Convergence for derivatives

Let $\{f_n\}$ be a sequence of differntible functions $f_n:\mathbb{R} \to \mathbb{R}$. And suppose that $\{f_n\}$ converges uniformly with rate $\varphi$ to a function $f$. That is: $$\sup_x||f_n(x) - ...
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241 views

determining the pointwise limit of a function

Consider $f_n : [0, 2] \to \mathbb{R}$ given by $$f_n(x) = \left\{\begin{array}{l} n^3 x^2, & 0 < x < 1/n; \\ n^3\left(x - \frac 2 n\right)^2, & 1/n \le x < 2/n; \\ 0, & ...
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2answers
50 views

Showing convergence with complex numbers

I would like to show that if $|a-b|<\delta$ then $|e^{a}-e^b|<\epsilon$. Where $a$ is complex and $b$ is real. In essence if the difference between a and b is small then the difference between ...
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1answer
30 views

Convergence of $a_n= (7/10+7/10i)^n$

I am new to real analysis and want to prove that $$a_n=\Bigg(\frac{7}{10}(1+i)\Bigg)^n$$ is convergent for $n\rightarrow \infty$. This is what I have done: $a_n= (7/10+7/10i)^n \\ = ...
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1answer
99 views

$L^1$ convergence of PDFs vs $L^2$ convergence of CDFs

Let $f_n$ denote a sequence of PDFs, and $F_n$ denote the corresponding sequence of CDFs. Given $L^1$ convergence of the PDFs to some PDF $f$, $$\int_\mathbb{R} |f_n(x) -f(x)| dx \rightarrow 0$$ ...
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2answers
37 views

A net in $\mathbb{R}$

Let $\{x_j\}_{j\in J}\subset \mathbb{R}$ be a net, $J$ is a directed set. If $\{x_j\}_{j\in J}$ does not converge to 0, then there is a subnet$\{x_b\}_{b\in B}$, $B$ is a directed set, that ...
2
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2answers
70 views

Is the series $\sum_{n=1}^\infty \frac{1}{1+n/2}$ convergent?

Is $$ \sum_{n=1}^{\infty} \frac1 {1+\frac{n}{2}} $$ convergent? I tried using the comparison test, but all I get is that it is inferior to the harmonic series, which is divergent.