91Ó°ÊÓ

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\arctan n}{n^{2}} $$

Use the limit comparison test to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{2}{3+\sqrt{n}} $$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{1}{n \sqrt{\ln n}} $$

Use a power series representation obtained in this section to find a power series representation for \(f(x)\). $$f(x)=\sinh (-5 x)$$

If the series is positive-term, determine whether it is convergent or divergent; if the series contains negative terms, determine whether it is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n-1} \frac{\sqrt{n}}{n+1} $$

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{2}{n^{3}+e^{n}}\)

Use the limit comparison test to determine whether the series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{1}{\sqrt{4 n^{3}-5 n}} $$

Find Maclaurin's formula with remainder for the given \(f(x)\) and \(n\). $$ f(x)=e^{2 x}, \quad n=5 $$

Find the first three terms of the Taylor series for \(f(x)\) at \(c\). \(f(x)=\sec x ; \quad c=\pi / 3\)

The overbar indicates that the digits underneath repeat indefinitely. Express the repeating decimal as a series, and find the rational number it represents. $$ 3.2 \overline{394} $$