91Ó°ÊÓ

a. Assuming that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant, show that $$ \lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0 $$ if \(c\) is any positive constant. b. Prove that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant. (Hint: If \(\varepsilon=0.001\) and \(c=0.04,\) how large should \(N\) be to ensure that \(\left|1 / n^{c}-0\right| < \varepsilon\) if \(n > N ?\) )

$$\text { Prove that } \lim _{n \rightarrow \infty} \sqrt[n]{n}=1$$

Prove that \(\lim _{n \rightarrow \infty} x^{1 / n}=1,(x > 0)\)

Determine whether the sequence is monotonic and whether it is bounded. $$a_{n}=\frac{3 n+1}{n+1}$$

Determine whether the sequence is monotonic and whether it is bounded. $$a_{n}=\frac{(2 n+3) !}{(n+1) !}$$

Determine whether the sequence is monotonic and whether it is bounded. $$a_{n}=\frac{2^{n} 3^{n}}{n !}$$

Determine whether the sequence is monotonic and whether it is bounded. $$a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}}$$

Determine whether the sequence is monotonic and whether it is bounded. Determine whether the sequence is monotonic, whether it is bounded, and whether it converges. $$a_{n}=1-\frac{1}{n}$$

Determine whether the sequence is monotonic, whether it is bounded, and whether it converges. $$a_{n}=n-\frac{1}{n}$$

Determine whether the sequence is monotonic, whether it is bounded, and whether it converges. $$a_{n}=\frac{2^{n}-1}{2^{n}}$$