Sequences

Exercise

1.

Find the limit of the following sequences:

(a)

$${\{a_{n}\} = \{n^4 - 3n^3\}}$$

(b)

$${\{a_{n}\} = \{\frac{3n^{2}+5}{4n^{3} - 1}\}}$$

(c)

$${\{a_{n}\} = \{\frac{1 - n}{n - \sqrt{n}}\}}$$

(d)

$${\{a_{n}\} = \{\frac{n(n+2)}{n+1} - \frac{n^{3}}{n^{2}+1}\}}$$

(e)

$${\{a_{n}\} = \{\sqrt{n+1} - \sqrt{n}\}}$$

2.

Show that for $a > 0$, $${\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$$ ．

3.

Find the limit of the following functions using $${\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1}$$.

(a)

For $a > b > 0$， $${\lim_{n \rightarrow \infty} (a^n + b^n)^{\frac{1}{n}}}$$

(b)

$${\{a_{n}\} = \{(1+2^{n}+3^{n})^{\frac{1}{n}}\}}$$