91Ó°ÊÓ

Evaluate the series or state that it diverges. $$\sum_{k=1}^{\infty}\left[\sin ^{-1}(1 / k)-\sin ^{-1}(1 /(k+1))\right]$$

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0$$

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{3 n^{2}}{4 n^{2}+1}=\frac{3}{4}$$

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty} \frac{\ln k}{k^{p}}$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Evaluate the series or state that it diverges. $$\sum_{k=1}^{\infty} \frac{(-2)^{k}}{3^{k+1}}$$

Evaluate the series or state that it diverges. $$\sum_{k=1}^{\infty} \frac{\pi^{k}}{e^{k+1}}$$

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty} \frac{1}{k \ln k(\ln \ln k)^{p}}$$

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} b^{-n}=0, \text { for } b > 1$$

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{c n}{b n+1}=\frac{c}{b}, \text { for real numbers } c > 0 \text { and } b > 0$$