91Ó°ÊÓ

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=0}^{\infty} \frac{e^{n}}{e^{n}+n} $$

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-5)^{-n} $$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=\sqrt{x+1}, \quad a=0\)

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

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}$$

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? $$ \sum_{n=1}^{\infty} \frac{1}{2 \cdot 4 \cdot 6 \cdots(2 n)} x^{n} $$

Find the first three nonzero terms of the Maclaurin series for each function and the values of \(x\) for which the series converges absolutely. \(f(x)=\cos x-(2 /(1-x))\)

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

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-100)^{n}}{n !} $$

Use series to evaluate the limits. \begin{equation} \lim _{y \rightarrow 0} \frac{y-\tan ^{-1} y}{y^{3}} \end{equation}