Question

Ce formula trebuie folosita pentru rezolvarea exercitiilor de genul acesta?

Answer 1

$\text{Se folosesc urmatoarele formule:}$

$\lim\limits_{n\to a}\left(1+\dfrac{1}{u(n)}\right)^{u(n)} = e,\quad \text{cand }u(n) \to \infty$

$e^{i\pi}+1 = 0 \Rightarrow e^{i\pi} = -1\Big|\sqrt{}\Rightarrow \sqrt{e^{i\pi}}=\sqrt{-1}\Rightarrow e^{\dfrac{i\pi}{2}} = i$

$4.\,\,\,\lim\limits_{n\to \infty}\left(1+i\dfrac{\pi}{2n}\right)^n = \lim\limits_{n\to \infty}\left(\left(1+\dfrac{1}{\dfrac{2}{i\pi}\cdot n}\right)^{\dfrac{2}{i\pi}\cdot n}\right)^{\dfrac{i\pi}{2}} = \\ =e^{\dfrac{i\pi}{2}}=\boxed{i}$

$5.\,\,\,\lim\limits_{n\to \infty}\left(1+\dfrac{1+\pi i}{n}\right)^n = \lim\limits_{n\to \infty}\left(\left(1+\dfrac{1}{\dfrac{n}{1+\pi i}}\right)^{\dfrac{n}{1+\pi i}}\right)^{1+\pi i}=$

$=e^{1+\pi i} = e\cdot e^{\pi i} = e\cdot (-1) = \boxed{-e}$