91Ó°ÊÓ

Test the series for convergence or divergence. $$\sum_{k=1}^{\infty} \frac{5^{k}}{3^{k}+4^{k}}$$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ \left\\{\frac{e^{n}+e^{-n}}{e^{2 n}-1}\right\\} $$

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. \(f(x)=e^{x}+e^{2 x}\)

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \sin \left(\frac{1}{n}\right)$$

For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? $$\begin{array}{ll}{\text { (a) } \sum_{n=1}^{\infty} \frac{1}{n^{3}}} & {\text { (b) } \sum_{n=1}^{\infty} \frac{n}{2^{n}}} \\ {\text { (c) } \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}} & {\text { (d) } \sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{2}}}\end{array}$$

Is the 50 th partial sum \(s_{50}\) of the alternating series \(\sum_{n=1}^{\infty}(-1)^{n-1} / n\) an overestimate or an underestimate of the total sum? Explain.

\(21-34\) Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \arctan n$$

\(21-34\) Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty}\left(\frac{3}{5^{n}}+\frac{2}{n}\right)$$

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{1+1 / n}}$$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ \left\\{\frac{\ln n}{\ln 2 n}\right\\} $$