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Results tagged with sequences-and-series
Search options answers only
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user 3272
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
5
votes
Longest coinciding pair of integer sequences known
By searching OEIS for the list 1,2,4,8,16, I found the following examples of sequences that start out as 1, 2, 4, 8, 16, but do not equal the sequence of powers of 2.
For $n \geq 1$, mark $n$ equa …
- 50.7k
43
votes
Accepted
Can a conditionally convergent series of vectors be rearranged to give any limit?
The Levy--Steinitz theorem says the set of all convergent rearrangements of a series of vectors, if nonempty, is an affine subspace of ${\mathbf R}^k$. There is an article on this by Peter Rosenthal …
- 50.7k
8
votes
Growth of a linear recurrent sequence
Besides the archimedean technique from Baker's theorem in another answer, this type of question can be handled both qualitatively and quantitatively using $p$-adic methods. This was illustrated, for …
- 50.7k
16
votes
Accepted
An Euler-proof that cannot be repaired?
This result of Euler is the last theorem in http://eulerarchive.maa.org/docs/originals/E072.pdf, and if you prefer English to Latin look at the last theorem in http://eulerarchive.maa.org/docs/transla …
- 50.7k
11
votes
A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$
This is asking for the value of an $L$-function of an even Dirichlet character $\chi$ at a positive even integer, and these have known values. It is analogous to the explicit expressions for the Riema …
- 50.7k
8
votes
Accepted
Series of reciprocals of smooth numbers
These are series of positive numbers, so they can be rearranged without affecting their values, whether or not they converge (look up Riemann rearrangement theorem).
Since $A^\otimes$ is the set of po …
- 50.7k
7
votes
Priming for the primes
Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information …
39
votes
Accepted
Parity of the multiplicative order of 2 modulo p
This problem was asked by Sierpiński in 1958 and answered by Hasse in the 1960s.
For each nonzero rational number $a$ (take $a \in \mathbf Z$ if you wish) and each prime $\ell$, let $S_{a,\ell}$ be th …
- 50.7k
9
votes
Accepted
Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}...
Here are analogous Tauberian theorems for power series and Dirichlet series that involve a condition of analytic continuation to a boundary point plus one extra condition that is necessary for the ser …
- 50.7k