91Ó°ÊÓ

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

Determine whether the sequence converges or diverges, and if it converges, find the limit. $$ \left\\{\left(1+\frac{1}{n}\right)^{n}\right\\} $$

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. $$ \sum_{n=1}^{\infty} \frac{e^{2 \pi}}{(2 n-1) !} $$

Find the interval of convergence of the power series. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{3^{n}}{n !}(x-4)^{n} $$

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

Find Maclaurin's formula with remainder for the given \(f(x)\) and \(n\). $$ f(x)=2 x^{4}-5 x^{3}, \quad n=4 \text { and } n=5 $$

Determine whether the series converges or diverges. $$ \sum_{n=1}^{x} \frac{n^{5}+4 n^{3}+1}{2 n^{8}+n^{4}+2} $$

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

Determine whether the sequence converges or diverges, and if it converges, find the limit. $$ \left\\{(-1)^{n} n^{3} 3^{-n}\right\\} $$

Find the interval of convergence of the power series. $$ \sum_{n=1}^{x}(-1)^{n} \frac{e^{n+1}}{n^{n}}(x-1)^{n} $$