Questions tagged [double-sequence]
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17
questions
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∈ \...
