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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A double series nature

here is defined a double series: $\displaystyle \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m} \frac{\cos(b\ln(2i))\cos(b\ln(2j))}{(2i)^{a}(2j)^{a}}-\frac{\cos(b\ln(2i-1))\cos(b\ln(2j-1))}{(2i-1)^{a}(2j-1)...
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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 ...
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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 ,...
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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 ...
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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 ...
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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^{\...
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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: $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\lvert a_ib_j\rvert$ ...
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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
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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\...
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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|=\...
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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} \...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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1 vote
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
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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 ...
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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 ...
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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 $(...
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4 votes
1 answer
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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 ...
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2 votes
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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 ...
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4 votes
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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,\...
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2 votes
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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 ...
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3 votes
4 answers
319 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}...
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3 votes
1 answer
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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,...
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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? ...
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2 votes
2 answers
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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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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 $i$?
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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 ...
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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}^{\...
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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 ...
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1 vote
1 answer
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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_{...
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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 ...
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2 votes
1 answer
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Sufficient condition for having finite double limit of a double sequence.

If $(s(n,m))$ is a double sequence such that (i) the iterated limit $\lim_{m \to \infty} (\lim_{n \to \infty }s(n,m))= a$, and (ii) the limit $\lim _{n \to \infty}s(n,m)$ exists uniformly for every $...
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6 votes
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What is the definition of double sequence $a_{mn}$ being convergent to $l$?

What is the definition of double sequence $a_{mn}$ being convergent to $l$? I have this definition. Definition: The double sequence $(a_{m,n})^∞_{m,n=1}$ is said to Converge to the real number $A∈ \...
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