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Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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Power method Shift and Convergence

Let $A\in \mathbb{R}^{{n}*{n}}$ be a symmetric matrix with eigenvalues satifying $\lambda_{1}>\lambda_{2}\ge\ldots \ge\lambda_{n-1}>\lambda_{n}$ and corresponding eigenvectors $x_{1},x_{2},\...
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Convergence of the exponentially-weighted moving average in integral form

Let $f:[0,+\infty)\to\mathbb{R}$ be continuous and let $\bar{f}\in\mathbb{R}$ be such that $f(t)\to\bar{f}$ as $t\to+\infty$. Does this suffice for \begin{equation} \int_{0}^{t}{\rm e}^{-\rho(t-s)}f(...
User129599's user avatar
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How to show that the limit of a sequence is not equal to some value?

In the second chapter of the book Understanding Analysis by Abbott, Example 2.2.6 proved that picking $N>\frac{1}{\varepsilon}$ suffices to prove the convergence of $\frac{n+1}{n}$ to the number $1$...
Nathan's user avatar
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Convergence of a sum sequence involving Beta function

I'm trying to decide whether following sequence converges or not: $ f_k = \sum\limits_{l=0}^{k-1}\left[ {{\Gamma{\left({q}+k+1\right)}}\over{\Gamma(k-l)\Gamma{\left({q}+l+2\right)}}} p^{l} \left(1-p\...
Iso's user avatar
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Convergence of Sequences with respect to Ultrafilters on the Natural numbers in Compact Hausdorff Topological Spaces

Let $(X,\mathcal T)$ be a topological space. Let $\mathcal F$ be an ultrafilter on $\mathbb N$. We say that sequence $(x_n)_{n\in\mathbb N}$ converges with respect to $\mathcal F$ to $x$, iff for all ...
Emancipatrix's user avatar
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Determine whether the series is convergent or divergent.

I've been working to determine if $$\sum_{x=2}^{\infty} \frac{1}{x^{\ln x}}$$ is convergent or divergent. $$\sum_{x=2}^{\infty} \frac{1}{x^{\ln x}} = \frac{1}{2^{\ln 2}} + \frac{1}{3^{\ln3}} + \frac{1}...
vvvvv's user avatar
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3 answers
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Convergence of $\sum\limits_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}x^n$

I was given the following question : Determine the radius of convergence of the following series : $\displaystyle\sum\limits_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}x^n$ I was able to determine using the ...
Johann Carl Friedrich Gauß's user avatar
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Why are limits of $L^p$ sequences defined almost surely? [closed]

I have heard it said that if a sequence of random variables $\{X_n\}$ converges in $ L^p $, then it converges to a limit $ X $ that is defined almost surely. I am trying to understand the precise ...
xy z's user avatar
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Continuum among converging sum of reciprocals of natural numbers sandwiched between consecutive terms

Define $u:=\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^{n+1}-1}\approx 0.6.\ $ Given $x\in \left[ u ,\ 1 \right]$, is there a (strictly increasing) sequence of positive integers $\left( a_{n} \right)_{...
Adam Rubinson's user avatar
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Trying to prove $(a^x)^y=a^{xy}$ using definition

In a lecture, we learned that $a^x$ is defined to be the limit of $a^{x_n}$ where $x_n$ is a series of rationals that converge to x. I was trying to prove $(a^x)^y=a^{xy}$ using solely this definition,...
PortyMart's user avatar
0 votes
2 answers
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Significance of Convergence Theorems in Analysis

We use theorems like Monotone Convergence, Dominated Convergence, Bounded Convergence theorems to show that when a sequence of measurable functions $f_n$ in a measure space $X$ converges to some ...
Grigor Hakobyan's user avatar
4 votes
1 answer
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Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$ \log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2} $$ where $\arg(z)$ is the standard branch of the ...
Max's user avatar
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$L_2$ convergence of bivariate function

I have the following problem: Let $X,Y$ be random variables with distributions $P_X,P_Y$ and $f_0$ be a map from the support of X,Y to the reals. I define a new function $\chi_0(y) = E_X[f_0(X,y)]$. ...
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Prove that the sequence defined by recurrence is periodic [duplicate]

Been stuck here for a while, can anybody help me? Thanks! Prove that the sequence defined recursively such that $$a_{n+1} = \frac{2}{2-a_n}$$ is periodic and has a period p = 4. We know that $a_1 \neq ...
Mircea Bodean's user avatar
1 vote
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A simple clarification on convergence of functions

Definition: $\lim_\limits{\large x\ \to\ x_0 \atop \large x\ \in\ E}f(x) = L$ iff for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\vert f(x) - L \vert \leq \epsilon$ for all $x ...
Community_Digest's user avatar
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1 answer
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How to quantify the "degree of convergence" so some number?

This might be a slightly silly question but let's say I have a finite sequence of $k$ numbers which might or might not converge to some constant value if $k$ were allowed to go to infinity. I'm only ...
ufghd34's user avatar
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We get $L^2$ convergence, but do we get a.s. convergence?

Assume you have a sequence of independent Bernoulli random variables $X_i$ each with probability $p_i$. Let $c_i$ be a sequence of real numbers and $ m,M$ be a real numbers such that $0 < m <c_i&...
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Convergence of linear functionals

Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of linear functionals in $\mathrm{BV}^*$, namely the dual of the space of functions with bounded variation. Suppose that $\ell_n$ converges to $\ell$ as $...
JayP's user avatar
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1 answer
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Problem E1 in Engels's "Problem Solving Strategies" [duplicate]

I am trying to solve problem E1 in Engels's book on the invariance principle. He includes a write-up of a solution, but it seems to me to be missing details. I'll paraphrase what he wrote. The setup ...
Cardinality's user avatar
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3 votes
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convergence of an infinite sum of exponentials

Suppose we have a series $$ f(x) = \sum_{i = 0}^\infty \exp(-a_i x) $$ with each $a_i \in \mathbb{R}$. Is the following true? Either the series diverges for all $x \ge 0$, or there exists some $R > ...
John Baez's user avatar
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0 votes
1 answer
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Convergence of Regula Falsi (false position) method

Let $f$ be a continuous function such that $f(a)f(b)<0$.Let $a_0=a$ and $b_0=b$ $$x_1:=a_0-\frac{b_0-a_0}{f(b_0)-f(a_0)}f(a_0)$$ If $f(a_0)f(x_1)<0$, set $a_1=a_0$ and $b_1=x_1$ If $f(x_1)f(b_0)&...
Arshdeep Sandhu's user avatar
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0 answers
33 views

Find a convergent series $\sum_{n=1}^{\infty} a_n$ such that $\sum_{n=1}^{\infty}1-\frac{\sin a_n}{a_n}$ diverges [duplicate]

I was given a homework assignment that goes like this: Let $a_n\neq0$ be a sequence such that $\sum_{n=1}^{\infty} a_n$ converges. Show that the series $\sum_{n=1}^{\infty} (1-\frac{\sin(a_n)}{a_n})$: ...
natitati's user avatar
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2 votes
2 answers
106 views

The relationship between $\sum_{n=1}^{\infty}\frac{b_n}{a_n}$ and $\sum_{n=1}^{\infty}\frac{b_1+\cdots +b_n}{a_1+\cdots+a_n}$

Let $a_n$ and $b_n$ be non-constant positive sequences. Given that the series $\sum_{n=1}^{\infty}\frac{b_n}{a_n}$ converges, I am curious about the convergence/divergence of the following series: $$ \...
Wing's user avatar
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Odd Function/Integral when evaluating a Moment

I am studying the Laplace (Double Exponential) distribution, and I have the following quote from Siegrist, which is quite direct, and not bothered about conditions being fulfilled: That the odd order ...
Starlight's user avatar
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1 answer
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Measurability of sets used to define convergence in measure

In Folland's Real Analysis: Modern Techniques and Their Applications, the following definition is given for a sequence of functions converging in measure. We say that a sequence $\{f_n\}$ of ...
Lightbulb's user avatar
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Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets)

To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.) Beginning of book quote End of book quote In the remark ...
Ulysse Keller's user avatar
2 votes
1 answer
109 views

Solving the non-linear recurrence $d_i + 2 i d_{i-1} = e_{i+1}$ (via generating functions)

I've been studying the recurrence $$d_i + 2i d_{i-1} = e_{i+1} =: (-2a)^{i+1}e^{-a^2} \quad\quad\quad a \in \mathbb{R}$$ attempting to solve for the sequence $(d_i)$. (The $d_0$ case is defined ...
Habeeb M's user avatar
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2 votes
2 answers
52 views

How do I get the positive series expansion of $(e^{\alpha x}-1)^{-1}$?

I couldn't figure out how the second line came out in the expression below. I basically have to expand $(e^{\alpha x}-1)^{-1}$ into a series, where $\alpha > 0$. My first try was $\beta \equiv e^{\...
xiver77's user avatar
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0 answers
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On continuous functions and convergent sequence

The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following. (a) If $\{x_n\}$ is convergent, then $\{f(x_n)\...
Nicholas Gray's user avatar
1 vote
2 answers
48 views

convergence of $\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p$

I want to find all $p\in \mathbb{R}$ such that the following series converges: $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p.$$ We know $$\frac{1}{n}-\sin\frac{1}{n}\leq \frac{1}{n^3}...
Ricci Ten's user avatar
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2 votes
1 answer
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Prove that $|X_n-X|^r\to 0 \Rightarrow E(|X_n|^r)\to E(|X|^r) $ (Vitali)

I am reading Vitali theorem here (statement page 4 and proof page 8) I am interested in the following part : Let $0<r<\infty$. Let $X_n$ and $X$ be $L^r(\Omega)$ random variables such that $X_n\...
Laurent Claessens's user avatar
1 vote
2 answers
78 views

Alternating series comparison test

Let's say I have two alternating series of terms, $(-1)^n A_n$ $(-1)^n B_n$ If I know (by for example Leibniz criteria) that one of the series converges / diverges, can I use comparison criteria to ...
Simeon Stefanović's user avatar
1 vote
1 answer
83 views

Ratio of two diverging integrals

Consider the ratio: $$ r = \frac{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a} x^2}{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a}} $$ For $a > 0$ we have $r = a$ because after ...
Uri Cohen's user avatar
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0 answers
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Show that Y_n converge in distribution to a law [duplicate]

Let be $\{U_i\}_{i\geq 1}$ a family of random variables independent and with the same distribution, $U[0,1]$ (uniform on [0,1]). Define $Y_n=\max_{1\leq i \leq n}\frac{U_i}{i}$ and show that $Y_n$ ...
Nicolas Rodriguez's user avatar
1 vote
1 answer
39 views

If a function is integrable and is of bounded variation, then must $\sum_{n\geq 1} f(n)$ converge?

Let $f:(0,\infty)\to [0,\infty)$ be a continuously differentiable function. Assume that $\int_0^{\infty} f(x)dx<\infty$ and that $\int_0^{\infty}|f'(x)|dx<\infty$. Claim: $\sum_{n=1}^{\infty}f(n)...
VShaw's user avatar
  • 353
2 votes
1 answer
77 views

Convergence of vector sequence when inner product converges to $0$

Let $d\in \mathbb N_{\ge 2}$, $(v_i)_{i\in \mathbb N}$ be a sequence of vectors in $\mathbb R^d$. If $\inf\limits_{N\in \mathbb N} \sup\limits_{i>j>N} v_i\cdot v_j =0$, does $v_i$ converge to $0$...
yummy's user avatar
  • 358
5 votes
1 answer
106 views

Is this proof correct for Putnam 1986 B4?

I have been practicing on some old Putnam questions, and I attempted to solve this 1986 Putnam B4: For a positive real number $r$ define $G(r)$ to be the minimum value of $|r-\sqrt{m^2+n^2}|$ for all ...
Riccardo Caiulo's user avatar
2 votes
3 answers
184 views

Convergence or divergence of $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$

If $a_1=a_2=1$ and $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$ How to prove convergence of the sequence and its limit or divergence? It is easy to see that the sequence is always positive and by that one ...
pie's user avatar
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0 votes
0 answers
40 views

Does $\int_1^\infty \frac{f_ng_n}{f_n^2+g_n} \, dx \to 0$ for $g_n < \frac{1}{x^3}$ and $f_n \to 0$? [duplicate]

Let $f_n, g_n : [1, \infty) \to (0, \infty)$ be sequences of measurable functions such that $$|g_n(x)| \leq \frac{1}{x^3} \hspace{0.5cm} \forall \, x \in [1, \infty) \hspace{0.5cm} \mathrm{and} \...
Grigor Hakobyan's user avatar
0 votes
1 answer
41 views

Show that convergence of mean square implies convergence of mean

Let $\{y_n\}$ be a sequence of real numbers and let $$\bar{y}_n = \frac{1}{n} \sum_{i=1}^n y_i \qquad s_n =\frac{1}{n}\sum_{i = 1}^n y_i^2$$ Suppose there exists a real number $s$ for which $$s_n \to ...
David's user avatar
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0 votes
0 answers
23 views

Understanding convergence rate of gradient descent

I am currently learning about gradient descent. For the convex case, I found this estimation in Nesterovs book: $f(x_k)-f^* \leq \frac{2L\|x_0-x^*\|^2}{k+4}$ Nesterov doesn't use the big o notation ...
user avatar
2 votes
0 answers
20 views

For what values of $k$ does the double integral of $\frac{1}{\|x_A - x_B\|^k}$ over intersecting surfaces converge?

I am working on a problem related to defining a potential energy function based on the distance between two intersecting surfaces in Euclidean space. Specifically, I am considering the following ...
cheng's user avatar
  • 21
2 votes
0 answers
34 views

Let $(X_i)_0^\infty$ be a sequence of i.i.d random variables S.T. $P(X_i > 0) > 0, P(X_i = i) > 0 \forall i$, what can we we say about $E[X_i]$?

Let $(X_i)_{i\in \mathbb{N}}$ be a sequence of independent identically distributed random variables such that $P(X_i > 0) > 0 \forall i$ and $P(X_i = i) > 0 \forall i$, can we say that $\...
Daan UTwente's user avatar
3 votes
1 answer
88 views

Existence of solutions in linear programming

If a linear programming problem "maximize $c^{\top} x$ with $Ax \leq b$, $x \geq 0$" is feasible (there is an $x$ satisfying the constraints) and bounded from above (there is a number $M$ ...
Mark's user avatar
  • 512
2 votes
1 answer
36 views

Convergence in $L^1_\text{loc}$ of weak derivates

Let $I$ an open inteval in $\mathbb{R}$. I'm looking for an example of a sequence of functions $(u_n)$ and a function $u$ in $I$ such that $u_n$ for all $n$ and $u$ has weak derivates $u_n'$ and $u'$ ...
matdlara's user avatar
  • 377
3 votes
1 answer
131 views

How to Prove Convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$?

For $\frac{\sin(\pi z)}{\pi z} =\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$, prove convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$. ...
pie's user avatar
  • 6,556
5 votes
1 answer
62 views

Equivalence classes and sequences

Suppose I'm working in $\mathbb{R}$ and I have the equivalence relation, $a\mathcal{R} b \leftrightarrow a \text{ mod } 10=b \text{ mod } 10$. Suppose now that I'm given a sequence $\{a_n\}_n\in \...
VAL's user avatar
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0 votes
0 answers
23 views

Proof verification: Show that if a series is conditionally convergent, then the series from its positive terms is divergent.

Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent. Please, help me to ...
user13's user avatar
  • 1,689
0 votes
1 answer
32 views

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$ The solution given in the book is as follows: Let the given integral be $\int_0^{\frac{\pi}{2}}f(x)...
Thomas Finley's user avatar
2 votes
3 answers
104 views

Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$

I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$. In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
Mariod's user avatar
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