잡담을 위한 소박한 장소

Enjoy this calculation and have a good day! Problem. For $\alpha > 0$, examine the following limit \begin{equation}\label{eqn:wts} \lim_{n\to\infty} e^{-\alpha\sqrt{n}} \sum_{k=0}^{n-1} 2^{-n-k} \binom{n-1+k}{k} \sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}. \end{equation} Solution. Let $A_n$ denote the formula inside the limit \eqref{eqn:wts}. By noting that the double summation is taken for ..

Today I want to deal with a problem which does not lie in analysis. It would be easy for those who are adapted to algebra, but for those with algebra trauma like me, it was rather a challenge. Problem. Let $m$ be an integer and $(u_n)$ be a sequence defined by \begin{equation}\label{eqn:recur} u_0 = 1, \quad u_1 = m \quad \text{and} \quad u_{n+2} = mu_{n+1} + u_n. \end{equation} Prove that for a..

Recently I was astonished by the following problem. Question.Let $f$ be a continuous function such that \begin{equation}\label{eqn:cond} \int_{-\infty}^{\infty} x^n f(x) \; dx = 0 \end{equation} for all $n = 0, 1, 2, \cdots$. Then is $f = 0$ everywhere? Answer. It might seem at the first glance that this is true. Surprisingly, however, this is not true. Let $$ g(x) = \exp\left(-x^2-\frac{1}{x^2}..