91Ó°ÊÓ

Find the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{1}{(-4)^{n}} x^{2 n+1} $$

Find Taylor's formula with remainder (11.45) for the given \(f(x), c,\) and \(n\). $$ f(x)=\tan ^{-1} x, \quad c=1, \quad n=2 $$

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. $$ \sum_{n=1}^{m}\left(n^{2}+9\right)(-2)^{1-n} $$

The expression is the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the first four terms and \(\lim _{n \rightarrow \infty} a_{n}\), if it exists. $$ 1+(-1)^{n+1} $$

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

Exer. 15-20: Approximate the integral to three decimal places, using the indicated exercise. $$ \int_{0}^{1 / 2} \sqrt{1+x^{3}} d x \quad \text { (Exercise 1) } $$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}+3}{(2 n-5)^{2}} $$

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

Find a Maclaurin series for \(f(x)\). (Do not verify that \(\left.\lim _{n \rightarrow \infty} R_{n}(x)=0 .\right)\) \(f(x)=\ln (3+x)\)

The expression is the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the first four terms and \(\lim _{n \rightarrow \infty} a_{n}\), if it exists. $$ \frac{n+1}{\sqrt{n}} $$