91Ó°ÊÓ

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n^{2}+1}$$

Use the Ratio Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}}$$

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} e^{-2 n}$$

Use the Direct Comparison Test to determine whether each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{n^{3 / 2}}$$

Use the Direct Comparison Test to determine whether each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n 3^{n}}$$

Each of Exercises gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\). $$a_{n}=\frac{2^{n}-1}{2^{n}}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$\frac{5}{1 \cdot 2}+\frac{5}{2 \cdot 3}+\frac{5}{3 \cdot 4}+\dots+\frac{5}{n(n+1)}+\cdots$$

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}+5}{n^{2}+4}$$

Find the first four nonzero terms of the Taylor series for the functions. $$\left(1-\frac{x}{3}\right)^{4}$$

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}$$