Questions tagged [double-sequence]

A double sequence is a map from $\mathbb{N}\times \mathbb{N}$ into a space; for instance, a double real sequence $(a_{ij})$ is a map $a:\mathbb{N}\times\mathbb{N}\to \mathbb{R}$. As with the single-variable case, the notation $\{a_{ij}\}$ is also common.

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Are convergent sequences closed under uniform convergence?

Setting: Let $Y$ be a metric space and let $a_{n,k}\in Y$ for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$. Suppose $a_{n,k}\to a_{\bullet, k}$ uniforly as $n\to\infty$. Suppose the sequences $\{a_{n,k}\...
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Limit Interchange for double sequence

I made the following claim and gave it a proof during my winter break. Given four hypothesis $X=\mathbb{N},Y$ is a metric space. $\{f_n\}$ uniformly converge to $f$. $\lim_{k\to \infty}f(k)$ exists. ...
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Convolution theorem of Laplace transform; Schiff

I'm reading Schiff's The Laplace Transform and I have some questions about the convolution theorem he proves on page 92 to 93. Theorem and proof Theorem 2.39 (Convolution Theorem). If $f$ and $g$ are ...
psie's user avatar
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Does diagonal double limit imply double limit?

If double indexed sequence $\{a_{m,n}\}$ has the same limit along each diagonal, i.e., $$ \lim_{m,n\to\infty\\ m/n\to c} a_{m,n} = a $$ for each $c\in(0,\infty)$, it is not necessarily true that the ...
North's user avatar
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Show $\sum\limits_{i=1}^n \sum\limits_{j=1}^n \cos(x_i - x_j) \geq 0$ for all real sequences $(x_i)_{1\leq i\leq n}$ [duplicate]

This inequality must be well known, and possibly easy to prove but I could not find it in the literature or here. Does anyone have a proof of $$ \forall n\in \mathbb{N}^*, \forall x_k \in \mathbb{R}, ...
Nathan Portland's user avatar
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A Diagonalization argument for Double limsup and /or liminf,

Let $\{F(k,n)\,:\:, n\geq 1, m\geq1\}$ be a double sequence family of extended real numbers $\Bbb R\cup\{-\infty,\infty\}$. I would like to prove the following statements 1- There is $(n_k)_k$, $n_k\...
Guy Fsone's user avatar
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How do I Derive a Mathematical Formula to calculate the number of eggs stacked on a crate?

A crate of eggs is stacked with eggs such that eggs are stacked on eggs in the crate until it is fully occupied and no egg can sit on another egg comfortably Derive a mathematical formula to determine ...
David Kunde's user avatar
2 votes
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Closed form of functional series

Watching this video it is "proven" that: If $\left |{x}\right |<1$ then $$\displaystyle\sum_{n=0}^\infty \dfrac{2^n x^{2^n}}{1+x^{2^n}} = \dfrac{x}{1-x}$$ The "proof" is as ...
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Iterated behaviour of Doubling map

In the dynamical system, there is one map called Doubling Map, defined as \begin{align} f:[0,1) & \rightarrow [0,1) & \\ f(x)&= \ \...
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Prove that if $\lim_{m,n\to\infty} a_{mn}$ exists and the iterated limits both exist, then all three limits must be equal. ("Understanding Analysis")

I am reading "Understanding Analysis 2nd Edition" by Stephen Abbott. Define $\lim_{m,n\to\infty} a_{mn}=a$ to mean that for all $\epsilon>0$ there exists an $N\in\mathbb{N}$ such that if ...
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Setwise limit of measures on a descending sequence of sets

This problem is from Royden's Real Analysis, (4-th e.d.), Chapter 18, Problem 66 (pp. 393). Problem Statement Let $(X, \mathcal{M})$ be a measurable space and $\{ \nu_n \}$ a sequence of finite ...
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double sequence question: convergence of the average over one index, when the sequence converges over another index

Let $\{a_{N,i}:i\geq 1, N\geq 1\}$ be a real valued double sequence, such that, $$ \lim_{N\rightarrow\infty} a_{N,i} = a_i, \quad\text{for each $i\geq 1$,} $$ where, $a_i\in \mathbb{R}$, for all $i\...
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Reference request for the convergences of different modes of double series: $\sum_{i,j}x_{i,j},\sum_{i}\sum_{j}x_{i, j},\sum_{j}\sum_{i}x_{i, j}$

I am looking for some reference text which goes over the convergence of different double series $\sum_{i,j}x_{i,j}$, $\sum_{i}\sum_{j}x_{i, j}$ and $\sum_{j}\sum_{i}x_{i, j}$. I have seen multiple ...
Epsilon Away's user avatar
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How do I solve the double summation $ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2}$?

Basically I'm stuck with this double summation. I want some help evaluating this summation. $$ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2} $$ Am I allowed to change the ...
Vaibhav C M's user avatar
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How do I prove the multivariable limit $\max\{ \frac{1}{\sqrt n} -\frac{\sqrt m}{ n},\frac{\sqrt n}{m} -\frac{1}{ \sqrt m}\}\to 0$?

While proving that $C([0,1])$ whith the norm $\|f\|=\sup_{t\in [0,1]}(t|f(t)|)$ is not complete the tutor used the following sequence: $x_n(t)=\cases{\frac{1}{\sqrt t}, t\in [\frac{1}{n},1]\\ \sqrt n ,...
some_math_guy's user avatar
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2 answers
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Is $\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma ?$

A friend proposed the following problem: $$\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma,$$ where $\gamma$ is the Euler-Mascheroni constant. The ...
Ali Shadhar's user avatar
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Interchange of order of limits of a double sequence

What are the sufficient conditions of interchanging two limits of a double sequence? I found some answers here: When can you switch the order of limits? However, I have the following example which ...
user2660120's user avatar
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Convergence in infinite double sequence

I was told in math class that with $\sum\limits_{n=1}\limits^{\infty}\lvert{a_{mn}}\rvert = b_m$ and $\sum\limits_{m=1}\limits^{\infty}b_m<\infty$, we can conclude that $\sum\limits_{m=1}\limits^{\...
flo's user avatar
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Let $s_{nn}=\sum_{i=1}^n\sum_{j=1}^na_ib_j$, prove that the limit of it is $AB$

THe problem first assumed that $\sum_{i=1}^{\infty}\lvert a_i\rvert=A,\sum_{j=1}^{\infty}\lvert b_j\rvert=B$ It then asked me to prove: $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\lvert a_ib_j\rvert$ ...
zony_miu's user avatar
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3 answers
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Calculating summation involving binomial coefficients [closed]

$$\sum_{0<=i<j<=7} \sum_{}\binom{7}{i} \binom{7}{j}$$ I'm not aware how to we proceed with such summations what do they represent so any way to teach me out this summation
imposter's user avatar
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Bounding a double sequence of integrals

I have a double sequence of probability density functions $\rho_{n,m}\in C^\infty(\mathbb R^d$). Suppose that for every test function $\varphi\in C_c^{\infty}(\mathbb R^d)$ $\lim_{m\to\infty}\lim_{n\...
tituf's user avatar
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5 votes
1 answer
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Proof verification: absolute convergence of double series

I would love some feedback on this proof. I've spent forever working on it. At this point my mind is too jumbled to realize if I can make it more efficient. I feel confident in it, but I will note the ...
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Does existence of both double limits imply boundedness?

Let $(a_{n,m})_{n\in\mathbb N,\,m\in \mathbb N}$ be a double sequence taking real values. We may consider different types of limit: $\lim_{n\to\infty,\,m\to\infty} a_{n,m} = l_1\in\mathbb R\,$ if for ...
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Similar formula for difference between partial double sums similar to that of single partial sum?

Is there a double sum formula for $$\left|\sum_{m=1}^{m_1}\sum_{n=1}^{n_1}x_{mn}-\sum_{m=1}^{m_2}\sum_{n=1}^{n_2}x_{mn}\right|$$ similar to how $$\left|\sum_{n=1}^{n_1}x_n-\sum_{n=1}^{n_2}x_n\right|=\...
c_gnar's user avatar
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2 answers
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bounded double sequence [closed]

Let $(a_{n,m})_{n,m\in\mathbb N}$ be a real double sequence. Suppose that $$ \forall m\in\mathbb N\quad \exists \lim_{n\to\infty} a_{n,m} = \lambda_m \in \mathbb R \,,$$ $$ \exists \lim_{m\to\infty} \...
tituf's user avatar
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Convergence of double summation series under diagonalization [closed]

I am stuck at 2.8.7b. The author suggested the strategy in the proof of theorem 2.8.1. But I can't see how the strategy can be deployed. As attached, the proof of theorem 2.8.1 involves 2 variables ...
zony_miu's user avatar
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2 votes
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Absolute convergence of a double matrix and of $\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{\frac{1}{2}+\epsilon}}$

According to Wikipedia the convergence of the right hand side of: $$\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{(\frac{1}{2}+\epsilon)}}$$ for $\epsilon$ an arbitrary ...
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4 votes
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General knowledge regarding double convergence of sequences (and series).

While proving a subspace of $\textit{L}^p(\mathbb{R}^n)$ is dense I stumbled upon a family of functions that was double indexed; this doesnt fall in our usual definition of sequence, it is merely a ...
Measure me's user avatar
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$\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) = \sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) = \sum_{(p,q)\in M} a_{pq}$?

I am reading "Lectures on Complex Function Theory" by Takaaki Nomura. There are the following two propositions in this book: Let $\sum_{(p,q)\in \mathbb{N}^2} a_{pq}$ be a double series of ...
tchappy ha's user avatar
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10 votes
2 answers
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Show $1+\frac{8q}{1-q}+\frac{16q^2}{1+q^2}+\frac{24q^3}{1-q^3}+\dots=1+\frac{8q}{(1-q)^2}+\frac{8q^2}{(1+q^2)^2}+\frac{8q^3}{(1-q^3)^2}+\dots$.

Show that $$1+\frac{8q}{1-q}+\frac{16q^2}{1+q^2}+\frac{24q^3}{1-q^3}+\dots=1+\frac{8q}{(1-q)^2}+\frac{8q^2}{(1+q^2)^2}+\frac{8q^3}{(1-q^3)^2}+\dots$$ where $|q|<1$ (q can be complex number). The ...
Charlie Chang's user avatar
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Prove absolutely convergent double series $\sum u_{\mu,\nu}$ when rearranged has the same limit.

Prove absolutely convergent double series $\sum u_{\mu,\nu}$ (which tends to S) when rearranged has the same limit. (1) First prove it is true for absolutely convergent series whose items are all non-...
Charlie Chang's user avatar
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Double series summation $S_{\mu, \nu}=\frac{\mu-\nu}{\mu+\nu}$

The double series $\sum u_{\mu, \nu}=\sum\frac{\mu-\nu}{\mu+\nu}$ is said to have sum by rows equaling -1, sum by columns equaling 1. But if we sum by rows, for each row, we have $\sum_{\nu=0}^\infty ...
Charlie Chang's user avatar
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1 answer
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For sample proof of formula $\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^s}=\zeta(s)\beta(s)-\zeta(2s),$ with s=2

I'm looking for a simple proof of this result only in case s = 2, $$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^s}=\zeta(s)\beta(s)-\zeta(2s),$$ there is proof here de T. Amdeberhan that I don't understand
Pascal's user avatar
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Limit of maximum of double indexed seqeunce

Suppose we have a uniformly bounded non-negative double indexed sequence $\{x_{m,n}\}_{m,n}$ such that $\forall m\in\mathbb N$ $\lim_{n\rightarrow\infty}x_{m,n}=1$. Then what can be said about the ...
RATul's user avatar
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Convergence of double sum and its rearrangement

Let $A=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij},a_{ij}\geq 0$ and $$\phi:\mathbb N \to\mathbb N\times\mathbb N$$ be any bijection. Now $B=\sum_{k=1}^{\infty}a_{\phi(k)}$. How $A$ and $B$ are ...
Cloud JR K's user avatar
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Changing the limits of a sum and combine them

could someone show me how to change the limits of a double summation $\sum_{i=1}^{n-1}\;\sum_{j = i+1}^n$ and combine them together to form $\sum_{i,j=1}^n$? i think we must incorporate the use of $(...
Ryantstrong 's user avatar
4 votes
1 answer
234 views

Finite sums involving Stirling number of first kind

I would like simplify the following doble sum $$ \sum_{k=m}^n\,s(n,k)\,x^k\sum_{s=m}^k\,(-1)^{k+s}\,s(k,s)\begin{pmatrix}s\\m\end{pmatrix}\,y^{s-m}$$ with $s(n,k)$ the Stirling numbers of first ...
popi's user avatar
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2 votes
1 answer
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Help in understanding a proof about infinite double summations

This is a problem from the book Understanding Analysis. I am struggling understanding to things. First I do not see how the order limit theorem is used, and second how they go from strictly less to ...
xxtensionxx's user avatar
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Evaluating $\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}\frac{\pi^{n+j}}{n!(j+n-1)^p}\frac{b_j}{j!}z^n$.

I know that the following functional series is absolutely convergent for every $z\in\mathbb{C}$ and $p‎‎>‎1$ $$\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}\frac{\pi^{n+j}}{n!(j+n-1)^p}\frac{b_j}{j!}z^n,\...
soodehMehboodi's user avatar
2 votes
0 answers
61 views

Limit superior and inferior of a set

If $S=\left\{\frac{1}{p}+\frac{1}{q}\mid p, q\in \mathbb{N}\right\}$ then $\varlimsup S-\varliminf S=?$ My attempt since derived set of S is$\left\{\frac{1}{n}|n\in \mathbb{N}\right\}\cup \{0\}$ and ...
user697748's user avatar
4 votes
4 answers
393 views

Check whether $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$ converges or NOT?

Check whether $$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$$ converges or NOT? My Try:- $\sum _{m=1}^{\infty }\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(m+n\right)^2}...
Truth_searcher's user avatar
3 votes
1 answer
667 views

Proof on Iterated Limits

This question is from a multi-step problem on iterated limits. First, we are given a doubly indexed array $a_{m,n}$ where $m, n$ $\epsilon$ $\mathbb{N}$. Define $\lim \limits_{m,n \to \infty} a_{m,...
nimo959's user avatar
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Four Dimensional Matrix

I have some multiple questions regarding 4D matrix. I understood that any element of a 4d matrix is itself a 2d matrix. Let $A$ and $B$ be two 4d matrices, then how to find $AB$, i.e. their product? ...
J. Yomcha's user avatar
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2 votes
2 answers
181 views

Double sum divergence [duplicate]

I have been trying to prove $$ \sum_{m=1}^\infty\sum_{n=1}^\infty \frac{1}{(m+n)^2}=\infty $$ I tried interchanging the sums, but the problem essentially same. I am also aware that: For $a_{m,n}\in [...
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0 votes
1 answer
273 views

Double series and iterated limits.

Can we construct a double sequence $\{x_{ik}\}$of non-negative numbers such that $\sum_k\sum_i x_{ik}<\infty$ while $\sum_k x_{ik}=\infty $ for each $i$?
NewB's user avatar
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0 votes
2 answers
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Harmonic Series sums to One?

I don't get it. How does $\sum_{j=1}^m\frac{1}{m}=1$? This looks like a harmonic series. I got this from brilliant.org, the website that trains students for AMC, AIME, Olympiad type of problems. This ...
Kenneth Dang's user avatar
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1 answer
59 views

Convergence of decreasing sequences on the infinite dimentional simplex

Let $N$ denote the set of natural numbers. Let $(x_{n,m})_{n,m \in N}$ be a double sequence of positive numbers that satisfies: 1) $x_{n+1,m} \leq x_{n,m}$, for every $m,n \in N$. 2) $\sum_{n=1}^{\...
MFG's user avatar
  • 78
1 vote
1 answer
65 views

No common terms in two sequences defined by linear quadratic recurrence relations

Let $x_n$ and $y_n$ be sequences such that $x_0=y_0=1, x_1=y_1=13$ and $$x_{n+2}=38x_{n+1}-x_n,$$ $$y_{n+2}=20y_{n+1}-y_n$$ for $n\ge0$. I want to show that there is no common terms when $n\ge2$. In ...
user680089's user avatar
1 vote
1 answer
458 views

Double supremum of double sequence

This is from "Measures, Integrals and Martingales" by R.L. Schilling, page 28, exercise 4.6ii. Prove that for any double sequence $\beta_{ij}, i, j \in \mathbb{N}$ of real numbers, we have $$ \sup_{...
baibo's user avatar
  • 619
3 votes
1 answer
414 views

lim$_{m , n \to \infty} S(n,m)$ exists but iterated limits do not.

$S(n,m)$ is a double sequence. Can anyone give me an example where lim$_{m , n \to \infty} S(n,m)$ exists but lim$_{n \to \infty}$( lim$_{m \to \infty} S(n,m)$) , lim$_{m \to \infty}$( lim$_{n ...
cmi's user avatar
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