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For each positive integer \(n,\) suppose \(s(n)\) is the sum of the digits of \(n.\) Find the smallest positive integer \(k\) such that

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\[ \large \lim\limits_{n \to \infty} \left( \frac{ \ln ( 1 + \frac{1}{n}) }{1} + \frac{ \ln (1 + \frac{2}{n})}{2} + \frac{ \ln(1 + \frac{3}{n})}{3} + \cdots + \frac{ \ln 2}{n} \right ) \]

A system consists of a set \(S\) of \(n\) independent components. A component \(a_i \in S\) is either working (with probability \(p_i\)), or has failed (with probability \(1-p_i\)), ...

Coupled oscillators are one of the most common physical systems in nature. For example, atoms in a crystal can be modeled as coupled oscillators. Moreover, the Taylor expansion of any ...

Let \(p_n\) be the \(n\)-th prime

Are there any polynomial functions \( F(x_1, x_2, \ldots, x_k) \) in \(k\) variables such that

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