Euler e

Exercise

1.

Determine whether the following sequences are bounded or not．

(a)

$${2,2^{2},2^{3},\cdots,2^{n},\cdots}$$

(b)

$a_{n}$ is approximated values of $\sqrt{2}$ taken to the $n$th decimal place．

2.

Find the limit of the following sequences $\{a_{n}\}$.

(a)

$${a_{1} = 1, a_{n+1} = \sqrt{3a_{n} + 4}}$$

(b)

$${a_{1} = 1, a_{2} = 2, a_{n+2} = \sqrt{a_{n+1}a_{n}}}$$

3.

Find the limit of the following sequences．

(a)

$${a_{n} = (1 - \frac{1}{n^2})^n}$$

(b)

$${a_n = (1 + \frac{2}{n})^n}$$

(c)

$${a_n = \frac{2^n}{n!}}$$

(d)

$${a_n = \frac{n!}{n^n}}$$