Euler e

Exercise

1.: Determine whether the following sequences are bounded or not．
(a): $\displaystyle{2,2^{2},2^{3},\cdots,2^{n},\cdots}$
(b): $a_{n}$ is approximated values of $\sqrt{2}$ taken to the th decimal place．
2.: Find the limit of the following sequences $\{a_{n}\}$ .
(a): $\displaystyle{a_{1} = 1, a_{n+1} = \sqrt{3a_{n} + 4}}$
(b): $\displaystyle{a_{1} = 1, a_{2} = 2, a_{n+2} = \sqrt{a_{n+1}a_{n}}}$
3.: Find the limit of the following sequences．
(a): $\displaystyle{a_{n} = (1 - \frac{1}{n^2})^n}$
(b): $\displaystyle{a_n = (1 + \frac{2}{n})^n}$
(c): $\displaystyle{a_n = \frac{2^n}{n!}}$
(d): $\displaystyle{a_n = \frac{n!}{n^n}}$