Convergence of sequences and different modes of convergence.

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2
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2answers
28 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. What can we say ...
0
votes
1answer
36 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 ...
-3
votes
1answer
26 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
votes
1answer
28 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 ...
0
votes
0answers
10 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 ...
1
vote
2answers
32 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
votes
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
58 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 ...
1
vote
1answer
18 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 ...
1
vote
4answers
52 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 ...
0
votes
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
votes
0answers
22 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 \}} + ... + ...
1
vote
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
vote
1answer
51 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
votes
1answer
27 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 ...
0
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0answers
19 views

Requesting help on understanding series [on hold]

Is the tangent of a positive convergent series still positive?
1
vote
4answers
51 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 ...
-1
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
vote
1answer
20 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?
0
votes
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 + ...
1
vote
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
votes
0answers
17 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
votes
1answer
43 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 ...
0
votes
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 ...
0
votes
0answers
25 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, ...
0
votes
0answers
18 views

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

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. ...
0
votes
1answer
13 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 ...
4
votes
3answers
98 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 $\pi^2/6$ ...
1
vote
0answers
60 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 ...
1
vote
1answer
30 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
votes
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 ...
1
vote
0answers
37 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 ...
1
vote
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 ...
1
vote
1answer
22 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
2
votes
0answers
25 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
4
votes
1answer
40 views

Convergence of the following sequence:

It could be exhaustion from the amount of work that I've done today, but I'd like to prove for myself that $$\lim_{n\to \infty} e^{-t\sqrt{n}}(1-\frac{t}{\sqrt{n}})^{-n}=e^{\frac{1}{2}t^2}$$ Here's ...
0
votes
2answers
41 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
0
votes
1answer
24 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
0
votes
0answers
34 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
0
votes
0answers
14 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
2answers
27 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
1
vote
1answer
24 views

Prove that the sequence $(b_n)$ converges

Prove that if $(a_n)$ converges and $|a_n - nb_n| < 2$ for all $n \in \mathbb N^+$ then $(b_n)$ converges. Is the following proof valid? Proof Since $(a_n)$ converges, $(a_n)$ must be bounded, ...
0
votes
0answers
16 views

show that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? [on hold]

Prove that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? Hint: let $f_n$ be the characteristic function of n-th interval in: ...
0
votes
1answer
58 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
1
vote
0answers
49 views

Prove the uniform convergence

Let $a_n$ be a monotonic sequence convergent to a. Let f : R $\to$ R be a continous and monotonic function. Then we define a series of functions as follows : $$f_n(x) := f(x+a_n)$$ Prove that the ...
0
votes
1answer
26 views

Euler method uniform convergence

I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's ...
1
vote
1answer
24 views

Almost sure convergence, arithmetic mean, variance

I am stuck proving that this sequence $$\sigma^2_n:= \frac{1}{n-1} \sum_{i=1}^n (X_i - \frac{X_1+...+X_n}{n})^2$$ is converegent almost surely to $D^2X_i = \sigma^2$. We assume that $X_1, X_2, X_3, ...
0
votes
0answers
21 views

Interchanging infinite double sum and expectation

Let $\xi_i$ be a sequence of independent and identically distributed standard normal random variables and consider sequences $\{b_i\}$ and $\{c_j\}$ such that $\sum_i b_i<\infty$ and $\sum_j ...
-1
votes
1answer
40 views

Give an example of a divergent and a convergent series such that the following holds: [closed]

I'm having trouble with this: I need to find an example of a divergent series $\sum_{n=1}^\infty a_n$ of positive numbers $a_n$ such that $lim_{n \rightarrow \infty }$ $a_{n + 1}/a_n$ = $lim_{n ...