Questions tagged [double-sequence]

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3
votes
1answer
147 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 ...
1
vote
1answer
49 views

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 ...
4
votes
0answers
250 views

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,\...
2
votes
0answers
39 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 ...
2
votes
4answers
127 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}...
2
votes
1answer
55 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,...
0
votes
0answers
45 views

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? ...
1
vote
2answers
105 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 [...
0
votes
1answer
31 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 $i$?
0
votes
2answers
40 views

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 ...
0
votes
1answer
41 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}^{\...
1
vote
1answer
34 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 ...
0
votes
1answer
135 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_{...
1
vote
0answers
30 views

Evaluate $\sum_{k=0}^{n}\sum_{j=0}^{t}(-1)^j{2n \choose 2k-j}^3=F(t)$

Given that $$\sum_{k=0}^{n}\sum_{j=0}^{3}(-1)^j{2n \choose 2k-j}^3=-3+(2n)^3+4{2n \choose n}{3n \choose n}$$ How would we evaluate the closed form for this sum? $$\sum_{k=0}^{n}\sum_{j=0}^{t}(-1)^j{...
3
votes
1answer
71 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 ...
1
vote
1answer
106 views

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 ...
3
votes
1answer
243 views

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∈ \...