Exam 2025 (A)

1a

\begin{matrix} Prove: (arccot x)^{'} = - \frac{1}{1 + x^{2}} \end{matrix}

\begin{matrix} Find lim_{x \to 0^{+}} \frac{1}{x} (\frac{1}{x^{x}} - 1) \end{matrix}

\begin{matrix} Let \sum a_{n} converges, \sum a_{n} = S \\ Let \sum b_{n} be a series obtained by changing order of \sum a_{n} \\ Prove or disprove: \sum b_{n} = S \end{matrix}

\begin{matrix} \sum_{n = 1}^{\infty} \frac{(2 n)!}{4^{n} (n!)^{2}} \end{matrix}

\begin{matrix} \sum_{n = 1}^{\infty} (\sqrt[n]{n} - 1) \end{matrix}

\begin{matrix} Let f be a differentiable function on R \\ Let {a_{n}}, {b_{n}} \subseteq R : a_{n} - b_{n} \to 0 \\ Prove or disprove: f (a_{n}) - f (b_{n}) \to 0 \end{matrix}

\begin{matrix} lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} \end{matrix}

\begin{matrix} a_{1} = 69420 \\ a_{n + 1} = \frac{1}{3} (2 a_{n} + \frac{5}{a_{n}^{2}}) \\ Find lim_{n \to \infty} a_{n} \end{matrix}