# Tag Info

### What is the limit of $a (n + 1) / a (n)$?

The value is close to $e$ but not. It's actually the positive real root of $p(t) := t^3 - 2t^2 + t - 8$. This is solvable via ACSV (see book by Pemantle and Wilson 2013). To summarize, the ...

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### "Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

This is to prove the conjecture \begin{equation*} x_n\sim\sqrt3\,n^{1/2} \tag{1}\label{1} \end{equation*} (as $n\to\infty$). (For all integers $n\ge1$,) we have \begin{equation*} h_n:=x_{n+1}-...
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### Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

See the paper "Le poisson n'a pas d'arêtes" by Thierry Bousch. This is a joke that was explained to me much later. The set is considered to resemble a fish. The French word arête means both bone and ...
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### Does anyone recognize these polynomials? Need to compute a riemann lebesgue type limit

Maple says $$\sum _{j=0}^{n}{\frac {{n\choose j} \left( -z \right) ^{j}}{j!}}={L}_n \left(z \right)$$ Laguerre polynomials. See https://en.wikipedia.org/wiki/Laguerre_polynomials and go down to "...
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### What does the abbreviation "p.p." mean in the context of convergence

This appears to be an abbreviation for presque partout, meaning almost everywhere. In the article you cite, reference is made to a paper of Hunt; the MathSciNet review for Hunt's paper (MR0236019) is ...
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### Subtracting the weak limit reduces the norm in the limit

The property you indicate is known as (strict) Opial’s Property (see https://en.m.wikipedia.org/wiki/Opial_property). It fails generally in reflexive spaces; in fact, it fails generally even for ...
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### Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$?

We will compute the generating function, and use the method described in section 2 of this paper. Let $F_{m,n}=F(m,n)$. Consider the generating function G(x,y)=\sum_{m=0}^\infty\sum_{n=0}^\infty F_{...
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### A term for sequences whose mean is defined?

The standard term is Cesàro summable, named after Ernesto Cesàro. Note that a convergent sequence is also Cesàro summable (with the same limit), but the converse does not always hold. Edit. I ...
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### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
Write it as $a_n(\alpha)$ to emphasize the dependence on $\alpha$. For any $\epsilon > 0$, $U(n,\epsilon) = \{\alpha: |a_n(\alpha)| < \epsilon\}$ is an open set containing $k/m$ for any integers ...