Infi-1 7

Division tools #theorem

\begin{matrix} L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | \\ \begin{matrix} 1. & L < 1 ⟹ a_{n} \to 0 \\ 2. & L > 1 ⟹ a_{n} \to \infty \\ 3. & L = 1 or ∄ L ⟹ Inconclusive \\ 4. & \sqrt[n]{| a_{n} |} \to L \end{matrix} \end{matrix}

\begin{matrix} lim_{n \to \infty} \sqrt[n]{n} \\ a_{n} = n \\ lim_{n \to \infty} \sqrt[n]{| a_{n} |} = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | = lim_{n \to \infty} | \frac{n + 1}{n} | = 1 \end{matrix}

\forall n \in N : a_{n} \leq a_{n + 1}

\forall n \in N : a_{n} < a_{n + 1}

\forall n \in N : a_{n + 1} \leq a_{n}

\forall n \in N : a_{n + 1} < a_{n}

Monotonic sequence always converges in the broadest sense

\begin{matrix} Let a_{n} be a monotonic sequence \\ \begin{matrix} 1. & Non-descending: \\ 1.1 & Top-limited: lim_{n \to \infty} a_{n} = s u p (a_{n}) \\ 1.2 & Not top-limited: lim_{n \to \infty} a_{n} = \infty \\ 2. & Non-ascending: \\ 2.1 & Bottom-limited: lim_{n \to \infty} a_{n} = i n f (a_{n}) \\ 2.2 & Not bottom-limited: lim_{n \to \infty} a_{n} = - \infty \end{matrix} \\ Proof for 1.1: \\ L = s u p (a_{n}) \\ a_{n} \leq L ⟹ \forall ε > 0 : a_{n} < L + ε \\ \forall ε > 0 \exists N : L - ε < a_{N} \\ ⟹ \forall n \geq N : L - ε < a_{N} \leq a_{n} \\ L - ε < a_{N} \leq a_{n} < L + ε ⟹ lim_{n \to \infty} a_{n} = L \\ Proof for 2.1: \\ L = i n f (a_{n}) \\ L \leq a_{n} ⟹ \forall ε > 0 : L - ε < a_{n} \\ \forall ε > 0 : \exists N : a_{N} < L + ε \\ ⟹ \forall n \geq N : a_{n} \leq a_{N} < L + ε \\ L - ε < a_{n} \leq a_{N} < L + ε ⟹ lim_{n \to \infty} a_{n} = L \end{matrix}

\begin{matrix} a_{n} = \sum_{k = n}^{3 n} \frac{1}{k} \\ a_{n} = \frac{1}{n} + \frac{1}{n + 1} + \dots + \frac{1}{3 n} \\ a_{n + 1} \leq a_{n} ⟺ \frac{1}{3 n + 1} + \frac{1}{3 n + 2} + \frac{1}{3 n + 3} \leq \frac{1}{n} \\ \frac{1}{3 n + 1} + \frac{1}{3 n + 2} + \frac{1}{3 n + 3} \leq \frac{1}{3 n} + \frac{1}{3 n} + \frac{1}{3 n} = \frac{1}{n} \\ ⟹ a_{n + 1} \leq a_{n} \\ ⟹ lim_{n \to \infty} a_{n} = i n f (a_{n}) \end{matrix}