Infi-1 6

Operations on infinites #lemma

\begin{matrix} \infty + \infty = \infty \\ C > 0, C \cdot \infty = \infty \\ C < 0, C \cdot \infty = - \infty \\ \dots \\ a_{n} \to \infty, b_{n} \to 0 \\ a_{n} \cdot b_{n} \to ? \\ \frac{0}{\infty}, \frac{\infty}{\infty}, \infty - \infty : Special cases \\ \frac{a_{n}}{b_{n}} \to ? \\ a_{n} = 17 n, b_{n} = 3 n \\ \frac{a_{n}}{b_{n}} = \frac{17 n}{3 n} = \frac{17}{3} \to \frac{17}{3} \\ a_{n} = 17 n, b_{n} = 3 n^{2} \\ \frac{a_{n}}{b_{n}} = \frac{17 n}{3 n^{2}} = \frac{17}{3 n} \to 0 \end{matrix}

\begin{matrix} a_{n}, b_{n}, c_{n} \\ {\begin{cases} a_{n} \leq b_{n} \leq c_{n} \\ a_{n} \to L, c_{n} \to L \end{cases} ⟹ b_{n} \to L \\ {\begin{cases} \forall ε > 0 \exists n_{ε_{a}} : \forall n > n_{ε_{a}} : | a_{n} - L | < ε \\ \forall ε > 0 \exists n_{ε_{c}} : \forall n > n_{ε_{c}} : | c_{n} - L | < ε \end{cases} \\ \forall ε > 0 \exists n_{ε} = m a x (n_{ε_{a}}, n_{ε_{c}}) : \forall n > n_{ε} : \\ L - ε < a_{n} \leq b_{n} \leq c_{n} < L + ε \\ ⟹ \forall ε \exists n_{ε} = m a x (n_{ε_{a}}, n_{ε_{c}}) : \forall n > n_{ε} : | b_{n} - L | < ε \\ ⟹ b_{n} \to L \end{matrix}

\begin{matrix} a_{n} \to L, a_{n} > 0 ⟹ L > 0 ? L \geq 0 ? \\ a_{n} = \frac{1}{n} \to 0 ⟹ L ≯ 0 \\ a_{n} \leq b_{n}, a_{n} \to L, b_{n} \to M ⟹ L \leq M ? \\ L - ε < a_{n} \leq b_{n} < M - ε \\ ⟹ L < M \\ a_{n} \to 0, m \leq b_{n} \leq M ⟹ a_{n} * b_{n} \to 0 ? \\ m \cdot a_{n} \leq b_{n} a_{n} \leq M \cdot a_{n} \\ m \cdot a_{n} \to m * 0 = 0 \\ M \cdot a_{n} \to M * 0 = 0 \\ ⟹ b_{n} a_{n} \to 0 \\ | a_{n} | \to 0 ⟺ a_{n} \to 0 \\ - ε < | a_{n} | < ε ⟹ - ε < a_{n} < ε \\ ⟹ | a_{n} | \to 0 ⟹ a_{n} \to 0 \\ - ε < a_{n} < ε ⟹ - ε < | a_{n} | < ε \\ ⟹ a_{n} \to 0 ⟹ | a_{n} | \to 0 \end{matrix}

\begin{matrix} a_{n} \to \infty, a_{n} \leq b_{n} ⟹ b_{n} \to \infty ? \\ \forall M > 0 \exists n_{M} : \forall n > n_{M} : b_{n} \geq a_{n} > M \\ ⟹ \forall M > 0 \exists n_{M} : \forall n > n_{M} : b_{n} > M \\ ⟹ b_{n} \to \infty \\ a_{n} \to - \infty, b_{n} \leq a_{n} ⟹ b_{n} \to - \infty \\ \forall M > 0 \exists n_{M} : \forall n > n_{M} : - b_{n} \geq - a_{n} > M \\ ⟹ \forall M > 0 \exists n_{M} : \forall n > n_{M} : - b_{n} > M \\ ⟹ b_{n} \to - \infty \end{matrix}

\begin{matrix} x > - 1 ⟹ (1 + x)^{n} \geq 1 + n x \end{matrix}

\begin{matrix} W > 1 ⟹ W^{n} \to \infty \\ W = 1 ⟹ W^{n} = 1 \to 1 \\ - 1 < W < 1 ⟹ W^{n} \to 0 \\ W \leq - 1 ⟹ W^{n} ↛ L, ↛ \infty, ↛ - \infty \end{matrix}

\begin{matrix} W > 1 \\ W^{n} = (1 + W - 1)^{n} \\ x = W - 1 \\ (1 + W - 1)^{n} \geq 1 + (W - 1) n \\ W > 1 ⟹ (W - 1) > 0 ⟹ 1 + (W - 1) n \to \infty \\ ⟹ W^{n} \to \infty \end{matrix}

\begin{matrix} - 1 < W < 1 \\ W = 0 ⟹ W^{n} = 0 \to 0 \\ | W |^{n} = | W^{n} | \\ | W^{n} | \to 0 ⟺ W^{n} \to 0 \\ | W^{n} | = {(\frac{1}{\frac{1}{| W |}})}^{n} \\ \frac{1}{| W |} > 1 ⟹ {(\frac{1}{| W |})}^{n} \to \infty \\ \frac{1}{{(\frac{1}{| W |})}^{n}} \to 0 \\ ⟹ | W^{n} | \to 0 ⟺ W^{n} \to 0 \end{matrix}