Convergence of sequences and different modes of convergence.

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2
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1answer
18 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 root 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
9 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
17 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 ...
1
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0answers
36 views

Weak convergence of a sequence of cdfs

Suppose $F_n \to F $ weakly, $ x \in c(F)$ and $ x_n $ is a real sequence converging to x. Prove that $F_n(x_n) \to F(x) $. Here $F_n$, $F$ are cdfs and $c(F)$-set of continuity points of $F$. I ...
3
votes
1answer
34 views

Convergence of a complex function

I need to proof if the following function is bounded and convergent. $f(n)=\left(\frac{10+in}{n^{2}+2in}\right)^{n}$ Status: This should be correct. Can anybody confirm this? I tried it with ...
0
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0answers
7 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
41 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 ...
0
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3answers
34 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
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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
26 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
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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$ ...
2
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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 $L^2$ ...
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$ ...
3
votes
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 ...
4
votes
3answers
65 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. ...
1
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1answer
187 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
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 ...
-3
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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?
0
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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 ...
1
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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
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}| < ...
11
votes
2answers
377 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
1
vote
1answer
59 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
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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 ...
1
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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 ...
0
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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 ...
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 ...
1
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2answers
67 views

Almost Sure Convergence

I was studying the various types of convergence and have studied almost sure convergence.I understand that almost sure convergence means $ Pr (\lim_{n\to \infty} X_n = X) = 1 $. I came across a ...
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 \}} + ... + ...
1
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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} ...
0
votes
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 ...
1
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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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0answers
19 views
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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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 ...
0
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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 + ...
0
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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
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0answers
29 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, ...
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, $P$, ...
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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 ...
0
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0answers
21 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. ...
4
votes
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 $\pi^2/6$ ...
0
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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 ...
4
votes
4answers
176 views

Convergence of $\left(1+\frac{1}{n}\right)^n$, given that $\left(1+\frac{1}{2^n}\right)^{2^n}$ converges

I know how to prove that the sequence $$\left(1+\frac{1}{2^n}\right)^{2^n}$$ has a limit. Can I use this knowledge to quickly get the fact that $$\left(1+\frac{1}{n}\right)^n$$ also has a limit?
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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
votes
1answer
39 views

Fourier series convergence question from big Rudin.

I am working on some problems from the 3rd edition of Rudin's "Real and Complex Analysis" and I'm stumped on proving the following part from question #19 of chapter 5. Suppose $\lambda_n/\log n \to ...
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 ...
0
votes
1answer
512 views

Convergence of Monotone sequences? example

An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...? According to the convergence of monotone sequences if a sequences is ...