91Ó°ÊÓ

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \sqrt[n]{n}(2 x+5)^{n} $$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \ln (1+2 x)$$

Use series to approximate the values of the integrals with an error of magnitude less than \(10^{-8}\) . \begin{equation} \int_{0}^{0.1} e^{-x^{2}} d x \end{equation}

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=2}^{\infty} \frac{\ln n}{\sqrt{n}} $$

In Exercises \(17-44\) , use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n !}{10^{n}}$$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0 . \overline{234}=0.234234234 \ldots $$

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1+\cos n}{n^{2}}\end{equation}

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(-3,-2,-1,0,1, \ldots\)

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,5,9,13,17, \dots\)

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty}\left(2+(-1)^{n}\right) \cdot(x+1)^{n-1} $$