91Ó°ÊÓ

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

Find the Maclaurin series for the functions. $$7 \cos (-x)$$

Determine if the geometric series converges or diverges. If a series converges, find its sum. $$\left(\frac{1}{8}\right)+\left(\frac{1}{8}\right)^{2}+\left(\frac{1}{8}\right)^{3}+\left(\frac{1}{8}\right)^{4}+\left(\frac{1}{8}\right)^{5}+\cdots$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$\cos ^{2} x\left(\text {Hint}: \cos ^{2} x=(1+\cos 2 x) / 2 .\right)$$

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

Which of the series in Exercises converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{3}{n+\sqrt{n}}$$

Find the Maclaurin series for the functions. $$18.5 \cos \pi x$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty} \frac{n x^{n}}{4^{n}\left(n^{2}+1\right)}$$

Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sqrt{n}}$$

Find a formula for the \(n\)th term of the sequence. $$-\frac{3}{2},-\frac{1}{6}, \frac{1}{12}, \frac{3}{20}, \frac{5}{30}, \dots$$