# Tagged Questions

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### Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$ where $p(n)$ is a polynomial equation. When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
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### Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$

I am currently interested in the following sequence: $$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275...$ with $C$ being the ...
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### Inverse problems for an asymptotic series which depends on a parameter?

I have the series $\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$, where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An ...
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Let $1<\alpha<\beta<3/2$. Set $$S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.$$ One can check that $S(n)$ is finite. My question is when $n\rightarrow ... 1answer 101 views ### Terminology for sequences/functions that approach each other What do I call two sequences$a, b$such that$\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions$f, g$such that$\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ... 1answer 422 views ### Limit of functions and asymptotic behaviour Let us consider a sequence$(p_l)_l$of polynomials on$[0,1]$that converge uniformely, as$l\to \infty$, to a function$f$defined on$[0,1]$. I denote the polynomials$p_l(t) = \sum_{k=0}^{m(l)} ...
Let $p_n$ be the nth prime and $p_L$ be closest to its square root: $$p_L^2 \approx p_n \approx x$$ Let $\sigma \in Z^+$ be a positive integer constant. Define the ...