91Ó°ÊÓ

Alternating Series Test Determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{\cos \pi k}{k^{2}}$$

Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge. $$\sum_{k=1}^{\infty} k^{-1 / 5}$$

Recurrence relations Write the first four terms of the sequence \(\left {a_{n}\right\\}\) defined by the following recurrence relations. $$a_{n+1}=3 a_{n}-12 ; a_{1}=10$$

Determine whether the following series converge. $$ \sum_{k=1}^{\infty}(-1)^{k} \frac{k^{11}+2 k^{5}+1}{4 k\left(k^{10}+1\right)} $$

The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. $$\sum_{k=1}^{\infty}\left(\frac{k}{k+1}\right)^{2 k^{2}}$$

Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{\sin (1 / k)}{k^{2}}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{k^{5}}{5^{k}}$$

Geometric series Evaluate each geometric series or state that it diverges. $$\sum_{k=0}^{\infty}\left(-\frac{9}{10}\right)^{k}$$

Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. $$\left\\{\frac{\sqrt{4 n^{4}+3 n}}{8 n^{2}+1}\right\\}$$

Recurrence relations Write the first four terms of the sequence \(\left {a_{n}\right\\}\) defined by the following recurrence relations. $$a_{n+1}=a_{n}^{2}-1 ; a_{1}=1$$