91Ó°ÊÓ

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

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. $$ \sum_{n=1}^{\infty} \frac{n+\cos n}{n^{3}+1} $$

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

Find Taylor's formula with remainder (11.45) for the given \(f(x), c,\) and \(n\). $$ f(x)=\ln \sin x, \quad c=\pi / 6, \quad n=3 $$

Use a power series representation obtained in this section to find a power series representation for \(f(x)\). $$f(x)=x^{2} e^{\left(x^{2}\right)}$$

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

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. $$ \frac{n+1}{\sqrt{n}} $$

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

Find Taylor's formula with remainder (11.45) for the given \(f(x), c,\) and \(n\). $$ f(x)=x e^{x}, \quad c=-1, \quad n=4 $$

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