Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
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questions with bounties
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Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets)
To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.)
Beginning of book quote
End of book quote
In the remark ...
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What is the number of integer sequences of length $T$ with fixed endpoints?
Let $T, n_1, n_2 \in \mathbb{Z}$ s.t. $T \geq 1$ be fixed. Consider the set
$$\mathcal{P}(0, T - 1, n_1, n_2) = \Big\{x:\{0, 1, \dots, T - 1\} \to \mathbb{Z} \space \Big| \space \big(x(0) = n_1\big) \...
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Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
- 812
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$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 814
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Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$
Let $X:=\{$ positive integers that contain the digit $2\}$
For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$
I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
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