Convergence of sequences and different modes of convergence.

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Finding a constant where an improper integral converges

I am trying to find the interval for the value $p$ where the following integral converges (log is the natural logarithm): $$\int_1^{\infty} \frac{\log{x}}{x^p} \,dx$$ So far I've used integration ...
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0answers
30 views

The convergence of series of independent random variables

Let $\{a_n\}$ be a sequence of complex numbers and let $\{A_n\}$ be a non-descreasing sequence of positive numbers, tending to infinity. We assume that $\sup\limits_{n\ge 1} ...
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3answers
33 views

Convergence test of $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$

Does S converge? (The answer says it converges) $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$ My attempt: Comparison test: $\ln (1+\frac{1}{k(k+2)}) \ln k \lt \ln 2 ...
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5answers
48 views

Show that $\{f_n\}_{n \geq 0}$ diverges in $(C[0,1], \|.\|)$.

On $C[0,1]$, we consider the usual norm $\| f\|=\sup\{|f(t)|: t \in [0,1]\}$. For $n\in N$, let $f_n \in C[0,1]$ define as $f_n(t)=t^{1/n}$. Show that $\{f_n\}_{n \geq 0}$ diverges in ($C[0,1]$, ...
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2answers
44 views

Contrapositive - Convergence of a sequence

I know that the convergence of a sequence $\{x_n\}_{n \geq0}$ in $\mathbb{R}$ is defined as for all $ \epsilon > 0$, there exists $N \in \mathbb{N}$ such that $|x_n-x|< \epsilon$ for all $n \geq ...
2
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1answer
23 views

Distance between the nullpoints of the series of derivatives of ln(x)/x

I plotted a function $f(x) = \frac{ln(x)}{x}$, and continued with $f'(x)$, $f''(x)$, $f'''(x)$. I noticed how the intersections between the functions and the x-axis seemed to be roughly equally ...
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0answers
48 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
2
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1answer
60 views

Is it true that $\lim\limits_{n\to\infty}{f(x_{n})}=\lim\limits_{n\to\infty}{f(y_{n})}$

Let $f: \mathbb{R}\rightarrow \mathbb{R}$. If for all sequence $x_n\rightarrow 0$, the sequence ${f(x_n)}$ is converges, can i say that it's true that for $x_n,y_n\rightarrow 0$, i will get: ...
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2answers
19 views

Convergence in probability of different uniform distribution R.V.s

A question in my book is as follows: Suppose $X_n$ is uniformly distributed on the set of points $\{1/n, 2/n, \ldots, 1 \}$. Does $X_n \xrightarrow{P} X$, where $X$ is $\mathcal{U}(0,1)$? The ...
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2answers
41 views

Convergence of Series of Functions $\sum_{n = 1}^{\infty}(n + 1)e^{1 - nx}$

I'm learning about series of functions and need some help with this problem: Given the series of function $\sum_{n = 1}^{\infty}(n + 1)e^{1 - nx}$ show that (i) it converges pointwise but ...
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1answer
44 views

Show Lebesgue integral $ \int_{0}^\infty \frac{\sin^2 {x}}{x^a}\,dx$ exist for $1 < a < 3$

Show that the Lebesgue-Integral $$ \int_{0}^\infty \frac{\sin^2 {x}}{x^a}\,dx$$ exist for $1 < a < 3$. This is the exercise I have to do. I know of one criteria, which says: if $f$ and $g ...
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0answers
18 views

Almost everywhere convergence plus convergence of integrals imply converence in L^1 [duplicate]

Consider non-negative measurable functions $f, f_n$ on a measure space $(X, \mathcal A, \mu)$. How does one show that $f_n \to f$ almost everywhere and $\int f_n d\mu \to \int f d \mu$ imply $f_n \to ...
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3answers
37 views

What is the significance of Convergence/Divergence of Series

There are many ways and mathematical methods to test the convergence or Divergence of a series. What does Convergence or divergence indicate practically in the real world? Why is the convergence of ...
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4answers
56 views

Prove the convergence of $\int_{0}^{1}(\ln(x))^2\,{\rm d}x$

Prove the convergence of $\int_{0}^{1}(\ln(x))^2\,{\rm d}x$ I know from plugging into a calculator that this is $2$. I'm just not sure how to show this. I want to come up with a function greater than ...
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2answers
40 views

Sum of infinite series with $i$ in ratio

I am trying to calculate the sum of an infinite geometric series. The problem is that in this series, '$i$' is part of the ratio. The equation is as follows, as best as I can produce it here: ...
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1answer
35 views

Complex numbers converge if their absolute values and arguments converge

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$. My proof: $z_n= |z_n|e^{i \arg(z_n)} ...
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2answers
162 views

Rate of Convergence vs Radius of Convergence

What is the difference between finding the 'rate of convergence' and the radius of convergence'? The question I am trying to solve here is to find the rate of convergence of the ratio of Fibonacci ...
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1answer
28 views

From pointwise convergence to uniform convergence

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $a\in \mathbb{R}$. Let $\{f_n(X, a)\}_n$ be a sequence of real-valued random ...
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2answers
46 views

Solving these types of integrals, using Monotone convergence theorem and Dominated convergence theorem.

I'm allowed to use these two theories and obviously the standard techniques when solving integrals. $$\lim_{n\to \infty } \int_{0}^{1}\frac{n^{\frac{1}{2}} x \ln x}{1+n^2x^2}dx$$ I did a similar ...
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3answers
65 views

Sum of a certain series [closed]

I'am having problems here. How to find the sum of this series? The factoral is confusing me. $$\sum_{n=2}^{\infty} \frac{(-1)^{n-1} 2^{n-1}}{(n-1)!}$$
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229 views

Can epsilon be a matrix?

Question In the following expression can $\epsilon$ be a matrix? $$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + ...
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3answers
54 views

Prove that $\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$ converges [closed]

Prove that the following power series converges: $$\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$$ I have tried using d'Alembert's ratio test however this was inconclusive. Anyone have any ideas?
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0answers
21 views

Is $\sum \frac{n+1}{b_{n+1}}$ irrational, when $b_1=2$ and $b_{k+1}=2^kb_k(b_k-1)+1$, $k\geq 1$?

Let the sequences of positive integers $$a_n=n$$ when $n\geq 1$, and $$b_{n+1}=2^nb_n(b_n-1)+1$$ for $n\geq 1$ taking $b_1=2$. I've computed with previous sequences to assert that satisfy the ...
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1answer
45 views

Study the convergence of this series of functions: $\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$

I tried to study the convergence of this series: $$\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$$ I started to study the pointwise convergence with the limit for $n ...
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0answers
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proof of a convergent subrow in every row in B using diagonal argument

full question I got a hint that for the compactness of M I need to show that every row in B has a convergent subrow (diagonal argument) but I don't know how to show this, does anybody know how to ...
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1answer
38 views

Series expansions with alternating series from endpoints: conjecture

A student of mine made the following two conjectures after working through examples of checking endpoints of series expansions, and although I feel that the conjectures are both false, I have not come ...
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1answer
27 views

Power series confusion when multiplying fractions.

I am stuck on the following question. check that the following sum from 0 to infinity converges using power series. sum of $$ 1/((n+(1/2))^2)$$ the next line of work is : $$4/((2n+1)^2)$$ I have ...
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2answers
32 views

Problems using D'alemberts Ratio test for convergence or divergence

The geometric series is as follows : $$n/2^n$$ I am using the ratio test therefore comparing : $$(n+1/2^{n+1})/ (n/2^n)$$ my next line of work is : $$(n+1/2^{n+1}) * (2^n/n)$$ however I am not ...
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0answers
30 views

Random Samples and Sample Variance Bound

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population. Show that: $$\max_{1 \leq i \leq n}|X_{i}-\bar{X}|<\frac{(n-1)}{\sqrt{n}}S$$ Where we have the sample variance ...
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1answer
20 views

Example of absolute convergent series divergent [duplicate]

I have learnt that in complete norm space any series that is absolute convergent is convergent. However, I am wondering is there any example of divergent series which is absolute convergent in that ...
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1answer
45 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
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5answers
60 views

Finding $\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$

Finding $$\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$$ I think if $$\int_{1}^{\infty}x^2 \frac{2^{x-1}}{3^x}dx$$ exists that this sequence is convergent, but I doubt that this integral is equal to ...
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3answers
32 views

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$ I am just not sure, I know if I use the absolute value test for convergence and root test it ...
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0answers
40 views

Define a sequence by two conditions and show this sequence converges and find its limit

Question: Define a sequence $(a_n)$ by two conditions $(a_1) = \sqrt{2}$ and $(a_n+1)= \sqrt {2+(a_n)}$ for $n \ge 1$. Show that this sequence converges and find its limit. (Suggestion: Show that the ...
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3answers
75 views

Does the integral converge? $\int_1^\infty \frac{ln(1+x)}{x^2}dx$

Does the integral converge? $$\int_1^\infty \frac{ln(1+x)}{x^2}dx$$ Well, I used integration by parts and got to $ln4$, which means it is clearly converges. but I want to try another approach as this ...
3
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1answer
106 views

Two nondecreasing sequences that bound each other

Question: Let ($a_n$) and ($b_n$) be two nondecreasing sequences with the property that, for each positive integer $n$, there are integers $p$ and $q$ such that $a_n \leq b_p$ and $b_n \leq a_q$. ...
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1answer
59 views

Use of directed sets in the definition of nets in topology

In topology, we use nets instead of sequences. The motivation is quite natural since the sequence is not "long" enough if the neighborhoods of some point "separate" too much. What I am confused ...
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6answers
52 views

Supremum proof simple

I got stuck on this problem and can't figure it out, I hope somebody can help me, I also wrote my attempt. Thanks in advance!! Question: Let $(a_n)$ be a convergent sequence in $\mathbb{R}$. $a_n ...
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1answer
20 views

Correct method of Proving Raabe's test?

I was wondering if my method of proof for Raabe's test was valid, since it is different from the normal method used with comparing to a sequence $\frac{1}{n^{p}}$ for some p > 1. Raabe's Test (As ...
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2answers
53 views

Which metric to use to make the sequence 1, 1.4, 1.414, 1.4142, .. converges in space Q?

In space Q, with the metric it inherits from R, the sequence 1, 1.4, 1.414, 1.4142, ... does not converge. Is there a way to change the metric to make it converge in Q?
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1answer
25 views

Non-linear systems convergence

Is there a way of being sure that simple iteration schemes, such as Gauss-Jacobi and Gauss-Seidel will converge for non-linear systems? I understand that for linear systems, the matrix A has to be ...
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1answer
40 views

Prove convergence of: $ \sum_{n=1}^\infty\frac{(-1)^n\cdot\sqrt{n}}{(n+1)\cdot2^n}\cdot(x-3)^n $

I would like to prove the convergence of series: $$ \sum_{n=1}^\infty\frac{(-1)^n\cdot\sqrt{n}}{(n+1)\cdot2^n}\cdot(x-3)^n $$ for every x $\in \mathbb{R}$. I am a bit lost on this one. I tried using ...
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Convergence of a sum of random variables with Bernoulli coefficient.

I present a problem which is connected with some of my previous questions. Suppose that $Y_t$ is a "regular enogh" (for example $Y_t=W_t$ with $W_t$ a Brownian motion) stochastic process with ...
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1answer
56 views

What about $\lim_{x\to 1}\left(\zeta(x)-\frac{1}{x^x-1}\right)=1+\gamma$?

When I type 1 in the box lim x to, and zeta(x)-1/(x^x-1) in the box Function, of this online calculator (Wolfram Alpha) one has as output $$\lim_{x\to ...
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1answer
24 views

Implications of convergence in probability

Consider two sequences of real-valued random variables $\{X_n\}_n$, $\{Z_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose that (1) $Z_n\in o_p(1)$, i.e. $Z_n$ ...
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0answers
16 views

Convergence of a sequence of Bernoulli variables.

For $\lambda\in(0,1)$ consider the following sequence of Bernoulli random variables $$ \mathbb{P}\left[B_n=1\right]=1-\frac{\lambda}{n},\quad \mathbb{P}\left[B_n=0\right]=\frac{\lambda}{n}. $$ Now ...
0
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1answer
41 views

Do these infinite series converge to a finite limit? [duplicate]

Now, I know that there is that remarkable result which finds that $$\sum_{n=1}^{\infty}n=-\frac{1}{12}$$ for $n\in\mathbb{N}$, under some kind of Cauchy limit. Are there any such convergences for ...
0
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0answers
25 views

Is the below assertion true? Why?

Let $n\in N$, $a_n$ be a real sequence such that $|a_{n+1}-a_n|\rightarrow 0$ as $n\rightarrow \infty$, then $a_n$ is convergent in $R$.
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0answers
36 views

Every nondecreasing function on [0,1] is the pointwise limit of a sequence of continuous functions.

Prove:Every nondecreasing function on $[0,1]$ is the pointwise limit of a sequence of continuous functions. I know every nondecreasing function can only have at most countable discontinuous point, ...
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1answer
22 views

Is this a sufficient condition for a.e. convergence?

Suppose one has a sequence $(f_{n})_{n \in \mathbf{N}}$ of real-valued, non-negative functions defined on a finite measure space $(X, \mu)$, with the following property: For every $n \in ...