91Ó°ÊÓ

Choose your test 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 or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}$$

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=3$$

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

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

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. $$\left\\{1.00001^{n}\right\\}$$

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

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. $$a_{n}=\frac{1}{4} a_{n-1}-3 ; a_{0}=1$$

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. $$\left\\{2^{n} 3^{-n}\right\\}$$

Use the properties of infinite series to evaluate the following series. $$\sum_{k=0}^{\infty} \frac{2-3^{k}}{6^{k}}$$