Convergence of sequences and different modes of convergence.

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Proof of convergence using comparison test [closed]

Is there a way to prove that these two sequences are divergent using comparison test? $\sum\limits_{n=1}^{∞}1/ln(n!)$ and $\sum\limits_{n=1}^{∞} µ(n)/n^2)$, where µ(n) is the number of digits of n.
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35 views

what is the difference between these two type of convergence and why people saying that that one of them is stronger than the another?

what is the difference between these two type of convergence and why people saying that that one of them is stronger than the another? In another word,how come point-wise convergence is weaker than ...
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2answers
26 views

let $\alpha >0$, and we define a sequence s.t $a_1 = \alpha $, and: $a_{n+1} = \sqrt{e^{a_n}-1}$, Prove that $a_n \rightarrow \infty$

let $\alpha >0$, and we define a sequence s.t $a_1 = \alpha $ and: $$ a_{n+1} = \sqrt{e^{a_n}-1} $$ Prove that $a_n \rightarrow \infty$. can I say that $e^x = 1 + ...
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1answer
82 views

$H^1$ convergence of eigenfunctions of Schrödinger operators

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
3
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1answer
34 views

Power series converges to $\{0\}$ or $ \mathbb{R}$

Given this theorem: If a power series $\sum_{n=0}^\infty a_n x^n$ converges at some point $x_0 \in \mathbb{R}$, then it converges absolutely for any $x$ satisfying $|x| < \left|x_0\right|$. ...
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148 views

Does $\sum a_n$ converge if $a_n = \sin( \sin (…( \sin(x))…)$

Does $\sum a_n$ converge if $a_n = \sin( \sin (\cdots( \sin(x))\cdots)$, $\sin$ being applied n times and $x \in (0, \pi/2)$? What about $\sum a_n^r$ for $r \in \mathbb{R^+}$?
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27 views

Convergence of the series - best criertion

What will be the best criterion to use to investigate convergence of the series (i do not need step by step explaination) $$\sum_{n=1}^\infty \frac{e^{\frac{1}{n}} }{n^{e}}$$
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41 views

Limit of exponential series

Find the following limit: $$\lim_{n \rightarrow \infty} \sqrt[n]{ \left( 3^n-2^n+1/2^n \right ) x^n }$$ How to calculate this limit? I am lookin for the solution that would involve use of squeeze ...
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3answers
116 views

Prove that $\sum_{n=1}^{\infty}{f\left({1\over n}\right)}$ converges absolutely.

Let $f:\Bbb{R}\to \Bbb{R}$ be continuously twice differentiable around $0$ such that $f(0)=f'(0)=0$. Prove that $\sum_{n=1}^{\infty}{f\left({1\over n}\right)}$ converges absolutely. What I did to ...
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0answers
32 views

Does this converge in probability?

I'm asked whether $X_{1}, X_{2}, ...$ converges in probability where $P\left(X_{n}=\frac{1}{n}\right)=1-\frac{1}{n^{2}}$ and $P\left(X_{n}=n\right)=\frac{1}{n^{2}}$. I think I have the solution, but ...
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2answers
54 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
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1answer
38 views

Finite Complements Topology and Convergent Sequences

Let $X$ be the set of natural numbers $\mathbb{N}$ together with $\mathcal{F}$, the finite complements topology. I've been asked to determine two different sequences $(x_{n}), (y_{n})$ such that: ...
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44 views

Beginner real analysis help

I have a little problem that I can't seem to get right. How do I show that $\sum_{n=1}^\infty \frac{4^n}{n! + 3^n}$ converges. I have tried using the ratio test on this, but I get something that just ...
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0answers
24 views

Proove the convergence of the Gauss-Seidel iterative method when the matrix is diagonally dominant

I'm reading about this proof here: However, I don't understand this part: "...from which (3.3) immediately follows" (in the upper half of page 3). Does it mean that: $||y|| \leq \gamma ||x||$ then ...
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1answer
38 views

Does $\sum (2n)!/(n!) $ converge p-adically

Does $\sum (2n)!/(n!) $ converge p-adically, I have worked out $v_p((2n)!) \leqslant 2n/(p-1) $ similarly $v_p((n)!) \leqslant n/(p-1) $ I want to prove this using the result that it converges ...
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1answer
79 views

convergence of a sequence

Be $(x_n)_n\ge 1 $ such that $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$ for every $n\ge 1$. Prove that the sequence $y_n=(2^n/x_n)_n \ge 1$ is convergent and find it s limit. Being positive and ...
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0answers
27 views

Universal localization argument for polynomially bounded functions of a stochastic proceess

As in many mathematical disciplines, many statements about stochastic processes are really easy to prove for bounded objects (sets, function, processes, etc.). This is why many proofs (e.g. for Ito's ...
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3answers
43 views

Convergence and Absolute Convergence of Arithmetic Mean of a sequence

Suppose $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n |x_i|$ exists. Does $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i$ exist? How about the converse? My thoughts: I guess for the sequence ...
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2answers
54 views

Does $\sum n $ converge p-adically?

Does $\sum n $ converge p-adically, I have worked out $v_p(n) \leqslant log(n)/log(p) $ not sure how to conclude from this I want to prove this using the result that it converges p-adically iff ...
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2answers
34 views

How do you prove that the following sequence is well defined and convergent

How do you prove that the following sequence is well defined and convergent? $a_{n+1}=\sqrt{2+\sqrt{a_n}}$ with $a_1=0$.
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1answer
29 views

Radius of Convergence of Power Series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$

What is the radius of the power series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$? Justify your answer. My steps toward a solution I can express $\tanh$ simpler as: \begin{align*} \tanh z ...
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1answer
80 views

Explanation for divergence of $\ln(x)$

$\ln(x)$ diverges as $x \to \infty$, yet the differential $\frac{1}{x}$ tends to $0$ as $x \to \infty$. To the naive mind, it seems that if the differential tends to $0$ then the original function ...
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1answer
26 views

Divergence of this series

I don't know how to solve this: Fix a positive real number $k$ and set $p=1/k$. For $n \ge 0$, consider the function $f_n : \mathbb{C} \rightarrow \mathbb{C}$ defined by $f_n(z)=k^n z^n$. Fix $z_0 ...
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2answers
207 views

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n(\sin(n)+2)}$ converge or diverge?

I was thinking about it and was stumped. Mathematica claims it converges.
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25 views

Sequence of continuous functions that converge pointwise to the Dirichlet function.

Prove that does not exist a sequence of continous functions that converge pointwise to the Dirichlet function $f:[0,1]\to\mathbb{R}$ defined as $f(x) $=0, if x is rational and $f(x) $=1, otherwise.
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1answer
19 views

Normal convergence of this function series

I don't know how to solve this task: Let $\mathbb{R}\_$ be the set of non-positive reel numbers and $U=\mathbb{C}\backslash\mathbb{R}\_$. For $n\ge 0$ consider the function $f_n=U \rightarrow ...
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59 views

Fourier series of $x^2$

I need to determine the fourier series of the following function $$f(x)=\left\{\begin{array}{cc}0, & x\in [-\pi, 0]\\ x^2 , & x\in (0, \pi] \end{array} \right. $$ and then show to which ...
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25 views

weak limit of $u_n(t,x) = \sin(nt)g(x)$ in $L^2(0,T;L^2)$

Let $u_n(t,x) = \sin(nt)g(x)$ for a function $g \in L^2(\Omega)$. We know that $\lVert u_n \rVert_{L^\infty(0,2\pi;L^2(\Omega))} < C$ because $\sin(nt) \leq 1$ and $C$ does not depend on $n$. So ...
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2answers
61 views

Arithmetic Mean convergence and concave transformations

Suppose $x_i \geq 0$ $\forall i$. If $\frac{1}{n}\sum_{i=1}^nx_i$ converges as $n\to\infty$, then what about $\frac{1}{n}\sum_{i=1}^nx_i^\delta$ where $0<\delta<1$? By Jensen's inequality we ...
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1answer
36 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
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1answer
28 views

Convergence of a sequence of a decreasing family of compact sets.

So I'm given a decreasing family of compact sets $(K_n)$ in $\mathbb{R}$ such that $K_1\supset K_2\supset K_3\supset \cdots$ and have to show that for a sequence $(a_n)$ such that $a_n \in K_n$ there ...
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1answer
27 views

How can I find the convergence radius for this series?

I want to find out the MacLaurin series of this function and find out for which $x$ it equals the original function: $f(x)=\frac{x}{1+3x^2}$ AFAIK I can use this equation: ...
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33 views

Newton Method convergence for different initial guesses

regarding the Newton method, we know that, if the function respects some constraints, it converges quadratically locally the solution. This is not proven for all the initial guesses, but only locally. ...
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3answers
34 views

Let $\{a_n\}$ be a sequence suppose there exists $0 <\lambda < 1$

Let $\{a_n\}$ be a sequence. Suppose there exists $0 < \lambda < 1$ such that $|a_{n+1} - a_n| \leq \lambda|a_n - a_{n-1}|$ for all $n >1$. Prove that $\{a_n\}$ converges. I seem to be ...
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1answer
27 views

Methods for Improving Convergence of a sequence of Partial Sums

I have the following sum: $$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$ Are there any methods that I can use to speed up the convergence of the sequence generated by ...
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2answers
208 views

Proving this sequence converges

Prove the sequence $A_n = \frac{1}{n+1}+ \frac{1}{n+2}+\ldots+ \frac{1}{2n}$ converges to a limit $a$ with $0 \leq a \leq 1$. Some help would be greatly appreciated been stuck on this for a day or so ...
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2answers
18 views

Consequence of the convergence of the series \sum_{n}\sin(nt)

It is known that if $\sum_{n}a_{n}<\infty$ then $\lim_{n\to\infty}a_{n}=0$. How to explain the situation of the series $\sum_{n}\sin(nt)$, for a fixed $0<t<2\pi$, since this series converges ...
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3answers
81 views

$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$

Let $a_i \ge 0$ $$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$$ I've got to this step $$\prod_{i=1}^\infty (1+a_i) = e^{\sum_{i=1}^\infty ...
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1answer
22 views

Find which z satisfy this equation

$$ z^3 = \sum_{n=1}^\infty (-1)^{n-1} \frac{1}{(i-1)^n} $$ How would i go about solving for $z$? I apologize for not being able to write the equation more readably.
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1answer
45 views

$\sum_{n=1}^{\infty} \left(1-n^{\frac{1}{n}}\right)^n$ converges?

$$\text{Does }\sum_{n=1}^{\infty} \left(1-n^{\frac{1}{n}}\right)^n \text{ converge?}$$ Using the ratio test we get $$\lim_{n\to\infty} ...
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1answer
18 views

(product of) uniformly convergent functions and pointwise convergence

Consider 2 sequences of real functions on $I \subset \Bbb R$: $f_n \to f$ and $g_n \to g$ uniformly. Need to prove that $f_ng_n \to fg$ pointwise on $I$ From definition I know that $\forall ...
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2answers
39 views

Using Dirichlet's test to prove $\sum_{n=1}^\infty a_n b_n$ converges

Suppose $\sum_{n=1}^\infty a_n = A$ exists and $\left(b_n\right)$ is a monotone sequence with limit $B$. $$\text{Prove }\sum_{n=1}^\infty a_n b_n \text{ converges?}$$ Can this be done using ...
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34 views

Prove if $x_n \leq k$ $\forall n \in N$ where $k$ is some constant integer, and $x_n \rightarrow x$, then $x \leq k$.

So we know that $|x_n - x| < \varepsilon$ for all $\varepsilon > 0$ for some $n \geq N$. Since $x_n \leq k$, I think we can say that $|x_n - x| \leq |k - x| < \varepsilon$ for all ...
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3answers
59 views

Find the sum of $\sum_{n=1}^{\infty}\frac{n}{x^n}$

The problem states: Wherever it converges, find the exact sum of $$\displaystyle\sum_{n=1}^{\infty}\frac{n}{x^n}$$ Using ratio test I know the series converges iff $|x|>1$, but have not idea how ...
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1answer
63 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
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22 views

$\lim\limits_{n\to \infty} \arctan(nx)$ and set-valued limit?

Consider the sequence $a_n(x)=\dfrac{2}{\pi}\arctan(nx)$. $(a_n)$ converges pointwise to $1$ if $x>0$, $-1$ if $x<0$ and $0$ if $x=0$. It does not converge uniformly as the limit function is ...
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2answers
32 views

Convergence of monotone decreasing series $\sum_{n=1}^\infty a_n < \infty \iff \sum_{n=1}^\infty 2^na_{2^n} < \infty$

Suppose $\{a_n\}$ is a monotone decreasing sequence of positive terms. Prove that $$\sum_{n=1}^\infty a_n \text{ converges } \iff \sum_{n=1}^\infty 2^na_{2^n} \text{ converges}$$ thought about the ...
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1answer
76 views

Uniform limit in definition of second order directional derivatives

If $f:E\rightarrow F$ is twice differentiable at $x\in E$, do we then have $$\lim_{h,k\rightarrow 0}\frac{A_x(h,k)-f''(x)(h)(k)}{\|h\|\|k\|}=0$$ where $A_x(h,k):=f(x+h+k)-f(x+h)-f(x+k)+f(x)$? This is ...
2
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0answers
50 views

Continuity of $t\mapsto \mathbb E X_t$

Is the following statement and its proof correct? Do you know this or related results and where I could read more about such things? Lemma: If $X$ is a right-continuous process, and the collection ...
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0answers
23 views

Random variables converge question

Let $X_1, X_2, \ldots$ be random variables that are independent and identically distributed. with $E[X_i] = 0, V[X_i] = 1$. Then there exist a random variable $Z$, that $(X_1 + X_2 + \cdots + ...