91Ó°ÊÓ

Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. $$a_{n}=\cot \left(\frac{n \pi}{2 n+2}\right)$$

Use the properties of infinite series to evaluate the following series. $$\sum_{k=2}^{\infty} 3 e^{-k}$$

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{5}=0.555 \ldots$$

Estimate the value of the following convergent series with an absolute error less than \(10^{-3}\). $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}$$

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

Use the properties of infinite series to evaluate the following series. $$\sum_{k=0}^{\infty}\left(3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right)$$

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

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. $$\left\\{0.2^{n}\right\\}$$

Consider the formulas for the following sequences. Using a calculator, make a table with at least ten terms and determine a plausible value for the limit of the sequence or state that the sequence diverges. $$a_{n}=\frac{5^{n}}{5^{n}+1} ; n=1,2,3, \dots$$

Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}$$