91Ó°ÊÓ

Find the first four nonzero terms in the Maclaurin series for the functions. $$\left(\tan ^{-1} x\right)^{2}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=1}^{\infty} \frac{1}{2 \cdot 4 \cdot 6 \cdot \cdots(2 n)} x^{n}$$

Use any method to determine whether the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n+1)(n+2)}{n !}$$

Converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=2}^{\infty} \frac{1}{n \sqrt{n^{2}-1}}$$

Use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=0}^{\infty} \frac{1}{n+4}$$

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

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

Use any method to determine whether the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} e^{-n}\left(n^{3}\right)$$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\frac{2 n+1}{1-3 \sqrt{n}}$$

Use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \frac{n}{n^{2}+3}$$