# 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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### 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 ...
67 views

### 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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### 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 ...
1 vote
52 views

### 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 ...
1 vote
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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 ...
106 views

### 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 ...
135 views

### 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 ...
36 views

### $\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) = \sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) = \sum_{(p,q)\in M} a_{pq}$?

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 ...
776 views

### 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 ...
1 vote
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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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### 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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### 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 ...
269 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∈ \...