91Ó°ÊÓ

Is it true that if \(\sum_{n=1}^{\infty} a_{n}\) is a divergent series of positive numbers then there is also a divergent series \(\sum_{n=1}^{\infty} b_{n}\) of positive numbers with \(b_{n}< a_{n}\) for every \(n ?\) Is there a "smallest" divergent series of positive numbers? Give reasons for your answers.

The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) In Exercises \(33-38\) , find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\) . $$ f(x)=1 / \sqrt{1-x^{2}} $$

Which of the series in Exercises \(11-44\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)^{n}}{(2 n)^{n}} $$

Converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{n !}{1000^{n}} $$

Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(27-38\) converge, and which diverge? Give reasons for your answers. $$ a_{1}=\frac{1}{3}, \quad a_{n+1}=\sqrt[n]{a_{n}} $$

In Exercises \(33-38,\) find the series' interval of convergence and, within this interval, the sum of the series as a function of \(x .\) $$ \sum_{n=0}^{\infty}\left(\frac{\sqrt{x}}{2}-1\right)^{n} $$

Which of the series in Exercises 1–36 converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{1}{1+2+3+\cdots+n} $$

Multiply the Maclaurin series for \(e^{x}\) and \(\sin x\) together to find the first five nonzero terms of the Maclaurin series for \(e^{x} \sin x .\)

In Exercises \(33-36\) , use series to estimate the integrals' values with an error of magnitude less than \(10^{-3} .\) (The answer section gives the integrals' values rounded to five decimal places.) $$ \int_{0}^{0.1} \frac{1}{\sqrt{1+x^{4}}} d x $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(-1)^{n+1}}{2 n-1} $$