91Ó°ÊÓ

Consider the following sequences. a. Find the first four terms of the sequence. b. Based on part (a) and the figure, determine a plausible limit of the sequence. $$a_{n}=2+2^{-n} ; n=1,2,3, \dots$$

Use the properties of infinite series to evaluate the following series. $$\sum_{k=1}^{\infty}\left(\frac{1}{3}\left(\frac{5}{6}\right)^{k}+\frac{3}{5}\left(\frac{7}{9}\right)^{k}\right)$$

Use the properties of infinite series to evaluate the following series. $$\sum_{k=0}^{\infty}\left(\frac{1}{2}(0.2)^{k}+\frac{3}{2}(0.8)^{k}\right)$$

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{027}=0.027027 \ldots$$

Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. $$\left\\{5(-1.01)^{n}\right\\}$$

Determine whether the following series converge absolutely, converge conditionally, or diverge. $$\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}$$

Consider the following sequences. a. Find the first four terms of the sequence. b. Based on part (a) and the figure, determine a plausible limit of the sequence. $$a_{n}=\frac{n^{2}}{n^{2}-1} ; n=2,3,4, \dots$$

Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{1}{5^{k}-1}$$

Determine whether the following series converge absolutely, converge conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}$$

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{12}=0.121212 \ldots$$