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Results tagged with sequences-and-series
Search options answers only
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user 4953
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.
3
votes
Asymptotics of a recurrence relation
Here is a sketch of the standard approach to a problem of this type; its expansion will probably of Lucia's solution.
If in Jeff Shallit's response you introduce the related monotone increasing seque …
- 13.5k
2
votes
Explicit formula for Euler zigzag numbers(Up/down numbers)
Dear Ross,
It looks that you don't really wish to see known formulae
for your zigzag numbers. Otherwise I don't understand why
you found my search insufficient.
The OEIS A000111 gives the formula
$$ …
- 13.5k
60
votes
What Are Some Naturally-Occurring High-Degree Polynomials?
From Fernando Rodriguez-Villegas' preprint:
"Chebyshev in his work on the distribution of prime numbers used the following fact
$$
u_n:=\frac{(30n)!n!}{(15n)!(10n)!(6n)!}\in\mathbb Z,
\qquad n = 0, 1 …
133
votes
Accepted
Convergence of $\sum(n^3\sin^2n)^{-1}$
As Robin Chapman mentions in his comment, the difficulty of investigating
the convergence of
$$
\sum_{n=1}^\infty\frac1{n^3\sin^2n}
$$
is due to lack of knowledge about the behavior of $|n\sin n|$ as …
- 13.5k
14
votes
What is the theoretical interest of finding closed-form solutions of infinite series?
I can recommend reading "Closed forms: what they are and why we care" by Jon Borwein and Richard Crandall, the article is to appear in Notices Amer. Math. Soc. 60 (2013).
Edit (Dec 2012). The paper h …
- 13.5k
3
votes
Rational numbers with dense orbits in [0,1] under iteration by f(x)=4x(1-x)
If I understand your question correctly, you ask about the trajectory of $\lbrace\xi\alpha^n\rbrace$ for $\xi\in\mathbb Q$ and $\alpha=2$.
Here is the abstract of [A. Dubickas, ON THE FRACTIONAL PART …
- 13.5k
16
votes
Accepted
Sum of subset of geometric series: a^2^n
Mahler proved in the 1930s that the values of $f(z)=\sum_{n=0}^\infty z^{d^n}$, $d>1$ is an integer, are transcendental for any algebraic $z$ satisfying $0<|z|<1$. A related problem of transcendence o …
- 13.5k
2
votes
Accepted
Relation between Legendre and Chebyshev polynomials
Both the Legendre and Chebyshev polynomials are particular cases of Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$. A general connection formula of the type $$P_n^{(\gamma,\delta)}(x)=\sum_{k=0}^nc_{n,k …
- 13.5k
6
votes
Is there a closed formula for the generating function of some trinomial coefficients?
What is "a way"? Of course, your question (in even more general form) was asked
centuries ago and gave rise to hypergeometric series, series of the form $\sum_n c_n$
with ratio $c_{n+1}/c_n$ being a r …
14
votes
Is this a rational function?
I would not argue with the irrationality of the function following from the functional equation relating $f(2z)$ to $f(z)$. In fact this function is a particular case of the so-called $q$-logarithm wh …
- 13.5k
12
votes
Reciprocals of Fibonacci numbers
To supplement Joseph's answer, I add my review MR2354148 on [C. Elsner, S. Shimomura and I. Shiokawa, Acta Arith. 130:1 (2007), 37--60].
Let $\lbrace F_n\rbrace _{n\ge0}$ and $\lbrace L_n\rbrace _{n\g …
- 13.5k