Question

Найдите предел последовательности

Answer 1

$\lim_{n \to \infty} a_n = \lim_{n \to \infty} (1+\frac{1}{n})^{2n}= \lim_{n \to \infty} [(1+\frac{1}{n})^n]^2= \lim_{n \to \infty}e^2=e^2$≈$7,389$

$\lim_{n \to \infty} b_n = \lim_{n \to \infty} (1+\frac{1}{2n})^{n}= \lim_{n \to \infty} (1+\frac{1}{2n})^{2n\cdot\frac{1}{2}}= \\= \lim_{n \to \infty} [(1+\frac{1}{2n})^{2n}]^{\frac{1}{2}}=\lim_{n \to \infty}e^{\frac{1}{2}}=e^{\frac{1}{2}}=\sqrt{e}$≈$1,649$

$\lim_{n \to \infty} c_n = \lim_{n \to \infty} (1-\frac{1}{n})^{n}= \lim_{n \to \infty} (1+\frac{1}{-n})^{-n\cdot(-1)}= \\= \lim_{n \to \infty} [(1+\frac{1}{-n})^{-n}]^{-1}=\lim_{n \to \infty}e^{-1}=e^{-1}=\frac{1}{e}$≈$0,368$