91Ó°ÊÓ

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. $$ \frac{2}{1 !}-\frac{2 \cdot 4}{2 !}+\cdots+(-1)^{n-1} \frac{2 \cdot 4 \cdots \cdots(2 n)}{n !}+\cdots $$

Find the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{n}{3^{2 n-1}}(x-1)^{2 n} $$

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

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

Determine whether the sequence converges or diverges, and if it converges, find the limit. $$ \left\\{\frac{\cos n}{n}\right\\} $$

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

Find the interval of convergence of the power series. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n 6^{n}}(2 x-1)^{n} $$

Determine whether the sequence converges or diverges, and if it converges, find the limit. $$ \left\\{\frac{e^{n}}{n^{4}}\right\\} $$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1+4^{n}}{1+3^{n}} $$

Determine whether the series converges or diverges. \(\sum_{n=1}^{x} \frac{n^{n}}{10^{n+1}}\)