Convergence of sequences and different modes of convergence.

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1answer
41 views

If $X_n\rightarrow X$ in mean square, then $\mathbb{E}(X_n)\rightarrow \mathbb{E}(X)$ [on hold]

How do I show that if $X_n\rightarrow X$ in mean square then $\mathbb{E}(X_n)\rightarrow \mathbb{E}(X)$ using the Cauchy-Schwarz inequality?
5
votes
5answers
67 views

Why is $f_n(x) = x^n$ not uniformly convergent on $(0, 1)$?

Definition of uniform convergence: For all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that $d(f_n(x), f(x)) < \epsilon$ for all $n > N \in \mathbb{N}$ and all $x \in (0,1)$. ...
1
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1answer
26 views

How to prove $\frac{nx^3}{1 + n^2 x^2}$ converges uniformly on $[1, \infty)$

I know this sequence of functions converges to $0$ pointwise, so I have to show that for all $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $d(\frac{nx^3}{1 + n^2 x^2}) < ...
0
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0answers
12 views

Commutation of Convolution, Restriction and Differentiation

Let $B$ be the open unit ball in $\mathbb R^n$ centered at zero and let $K=\bar{B}\cap (\mathbb R^{n-1}\times\{0\})$. Suppose you are given $u\in C^{1,\alpha}(B)$ such that $u|_K=f\in C^2(K)$. For a ...
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votes
3answers
31 views

Determing the radius of convergence of the following power series. [on hold]

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. The series is: $$\sum_{k=0}^\infty ...
2
votes
1answer
27 views

For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

I used the ratio test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the ...
1
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1answer
13 views

Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$

Assume that a sequence $(z_n)$ of complex numbers converges to a nonzero limit. Then Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$ I know I should ...
1
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2answers
18 views

Image of a convergent sequence in an increasing function

Suppose I have a function $f:[a,b) \to \mathbb{R}$ that is increasing on its domain and a sequence $a_n \subset [a,b)$ such that $a_n \to b$ and $f(a_n)\to l\in \mathbb{R}$. How would I go about ...
0
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1answer
14 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
1
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3answers
43 views

Limit function of $\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$

What is the limit of the sequence of functions$\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$? I think this just doesn't exist, since it goes to $\infty$ in $x=0$ and goes to $0$ everywhere ...
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3answers
35 views

Calculate the radius of convergence of $\sum \frac{\ln(1+n)}{1+n} (x-2)^n$

Calculate the radius of convergence of the following: $$ \sum \frac{\ln(1+n)}{1+n} (x-2)^n $$ Will you please help me figure out how to calculate: $$ \lim_{n\to \infty} \frac{\ln(2+n)}{2+n} ...
1
vote
1answer
30 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
1
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0answers
28 views

fn converges to f pointwise where all functions fn are bounded and f is unbounded: Is this example correct?

I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded. I have thought of an example where the ...
2
votes
0answers
24 views

Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
2
votes
2answers
31 views

Convergence: infinite series

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}, n\geq\text{some integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
0
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2answers
26 views

Sequence and series : Convergence

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} <= \frac{b_{n+1}}{b_n}, n>=\mathrm{some\space integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
4
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3answers
66 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
0
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1answer
39 views

How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?

How would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but ...
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votes
1answer
27 views

Define $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold? [on hold]

Let $c\in \mathbb{R},c>0$ and define the sequence $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold?
3
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1answer
42 views

Analyzing convergence of series with sine and cosine

Analyze the convergence of the following series: $$\displaystyle\sum\frac{\cos{n}}{\sqrt{n}+\cos{n}}$$ $$\displaystyle\sum\frac{\sin{n}}{\sqrt{n}+\cos{n}}$$ I tried to use the direct comparison test ...
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0answers
12 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
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2answers
35 views

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge?

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge? My try:As $n$ approaches zero, $\sec(\frac 1n)$ gets close to $\frac 1{1-0.5\frac 1{n^2}}=\frac{2n^2}{2n^2-1}=1+\frac ...
0
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1answer
43 views

Does $\frac{x}{n}$ converge uniformly on ℝ?

Does $x, \frac{x}{2}, \frac{x}{3}, \frac{x}{4}, \ldots$ converge uniformly on ℝ? I think that it does not since $\lim_{n\rightarrow+\infty} x/n = 0$. Then $|\frac{x}{n} - 0| = |\frac{x}{n}| < ...
1
vote
1answer
61 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
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1answer
19 views

Finding interval of convergence for complicated sum

I'm going through old exams for my Calc III course and came across a problem that I did not know how to do. The problem is: Find the interval of convergence of the series ...
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4answers
55 views

Prove that $\lim_{n\to \infty} n\cdot r^n=0$ where $(0\leq r <1)$ without using ratio test

$\lim_{n\to \infty} n\cdot r^n=0$, where $0\leq r <1$, can be obtained by vanishing condition (considering $\sum^{\infty}_{n=1}n\cdot r^n$, which converges, using ratio test). Is there a direct ...
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0answers
9 views

MathCAD symbolic solution convergence time (variables only) [on hold]

This question would more likely be dedicated to MathCAD users. Thank You. How long would you think a symbolic equation would take to converge for the solution of a variable in MathCAD ? Provided that ...
0
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0answers
25 views

Random variables, indicator variables, conditional probability

Let $\{X_n \}$ be a sequence of independent identiacally distributed random variables. Let $A, B \in \mathcal{B}(\mathbb{R})$ be such that $P(X_1 \in B) \neq 0$. Let $S_n:= 1_{\{X_1 \in B \}} + ... + ...
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0answers
25 views

Binary system, number of $1$s, almost sure convergence

Could you check if my solution is correct? For $x \in [0,1]$ let $S_n$ be the number of times $1$ occurs in the first $n$ digits of $x$'s binary representation. Show that $\lim _{n \to \infty} ...
1
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1answer
53 views

Limit comparison test

Since the series $\sum_{n=1}^\infty a_n$ is convergent, so $\lim_{n\to\infty} a_n=0.$ Consider $$\lim_{n\to\infty} {\left(\dfrac{{a_n}^{1/2}}{n}\right)\over{\left(\dfrac{1}{n^2}\right)}} $$ which is ...
0
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1answer
29 views

Stuck on finding where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally.

I have all the questions correct on my hw except for one: find where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally. Radius of Convergence I got 1 for this, by using the root test and finding ...
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0answers
19 views

Requesting help on understanding series [on hold]

Is the tangent of a positive convergent series still positive?
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4answers
52 views

Proof that $0.33333… = \frac{1}{3}$ using $\epsilon-N$ method

This proof is quite prevalent on the web, yet I struggle using this particular method. Wikipedia (http://en.wikipedia.org/wiki/Limit_of_a_sequence) tells us: We call $x$ the limit of the sequence ...
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votes
1answer
32 views

I have a feeling that these statements on my homework are true, but how would I prove it? [on hold]

If $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are both absolutely convergent series with all positive terms then $\sum_{n=1}^{\infty}a_n/b_n$ is absolutely convergent. If the power ...
1
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1answer
23 views

Uniform convergence and continous.

Let $(f_{n})$ be a sequence of functions. Is it possible that $(f_{n})$ converges uniformly where each functions (that is $f_{1},f_{2}, f_{3}\dots$) aren't necessarily continous?
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2answers
33 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
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1answer
11 views

Describing convergence/divergence of a complex sequence

Let (a$_n$)$_{n \in N}$ be a complex sequence and a $\in$ C. Show that the following statements are equivalent: $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ N $\forall$ n $\geq$ N : |a$_n$ - a| ...
3
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0answers
18 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
0
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1answer
45 views

Limit of a convergent serie

For a research project, after some manipulation I come up with a convergent serie that I have to prove its limit. The statement is the following: $ \lim_{n \rightarrow \infty } \displaystyle ...
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0answers
7 views

If $S_M$ denotes the measure of a submanifold, then $\frac 1{r^{n-1}n\omega_n}\int_{\partial B_r}u(x)\;dS_{\partial B_r}(x)\to u(y)$ for $r\to 0$

Let $S_M$ denote the "surface measure" of a submanifold $m$ $B_\varepsilon(y)$ denote the open ball around $y$ with radius $\varepsilon>0$ $\omega_n$ denote the volume of the $n$-dimensional unit ...
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0answers
30 views

Rate of Convergence of a Sequence Defined by a Function

In my notes, I have that if a sequence defined by a function, $x_{i+1} = f(x_i)$, converges to $c$ in the limit, i.e., $$\lim_{i\to\infty} x_i = c,$$ then the rate of convergence to this limit, ...
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0answers
21 views

What does it mean for a sequence of complex functions to converge uniformly? [closed]

Let E $\subset$ C. What does it mean for a sequence of complex functions with domain E to converge uniformly? Give an example of a sequence of functions with domain C that converges but not uniformly. ...
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1answer
14 views

How to show $[0, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, 2\pi)$ is sequentially separated?

Definition of separated: $E$ is separated if $A, B \neq \varnothing$, $A \cup B = E$, and there are not convergent sequences in $A$ that have limit points in $B$, and vice versa. Definition of ...
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3answers
104 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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0answers
63 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
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1answer
32 views

Newton Integral Convergence

please, I have a problem. I suppose it´s quite easy, however, I really don´t see what should I do with it. I should decide on convergence or divergence of this integral: $$\int_0^\infty ...
0
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1answer
16 views

A very simple question about multinomial distributions

Let's say you have a random vector $(x_1,\ldots,x_k)$ that has a multinomial distribution with parameters $n$ and $(p_1,\ldots,p_k)$. Suppose that we know $p_i>p_j$ for some $i,j$. Is it correct ...
2
votes
1answer
16 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
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0answers
40 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
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0answers
14 views

Multivariate Berry-Esseen/ Please help!

I've got a problem with understanding Berry-Esseen inequality for random vectors. You see, I keep coming across various forms of this theorem, all assuming a unit covariance matrix $I$, though it's ...