91Ó°ÊÓ

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

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. $$a_{n}=2+2^{-n} ; n=1,2,3, \ldots$$

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{k^{2}+2 k+1}{3 k^{2}+1}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3 / 2}}$$

Determine whether the following sequences converge or diverge and describe whether they do so monotonically or by oscillation. Give the limit when the sequence converges. $$\left\\{(-0.7)^{n}\right\\}$$

Determine whether the following sequences converge or diverge and describe whether they do so monotonically or by oscillation. Give the limit when the sequence converges. $$\left\\{(-1.01)^{n}\right\\}$$

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. $$a_{n}=\frac{n^{2}}{n^{2}-1} ; n=2,3,4, \dots$$

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

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). $$0 . \overline{027}=0.027027 \ldots$$