91Ó°ÊÓ

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. $$0.6+0.06+0.006+\cdots$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=0}^{\infty}\left(\frac{\tan ^{-1} k}{\pi}\right)^{k}$$

Choose your test Use the test of your choice to determine whether the following series converge. $$\frac{1}{2^{2}}+\frac{2}{3^{2}}+\frac{3}{4^{2}}+\cdots$$

Use the Ratio Test or the Root Test to determine the values of \(x\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}$$

Determine whether the following series converge absolutely, converge conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{\sin k}{3^{k}+4^{k}}$$

More sequences Find the limit of the following sequences or determine that the sequence diverges. $$a_{n}=\frac{e^{-n}}{2 \sin \left(e^{-n}\right)}$$

Series of squares Prove that if \(\sum a_{k}\) is a convergent series of positive terms, then the series \(\Sigma a_{k}^{2}\) also converges.

Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. $$\sum_{k=1}^{\infty} \frac{3^{k+2}}{5^{k}}$$

Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. $$4+0.9+0.09+0.009+\cdots$$

More sequences Find the limit of the following sequences or determine that the sequence diverges. $$\left\\{\frac{\tan ^{-1} n}{n}\right\\}$$