91Ó°ÊÓ

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{1+1 / n}}$$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{2^{1+1 / n} n^{1+1 / n}}$$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin \left(\frac{1}{n}\right)\right)$$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty}\left(1-\cos \left(\frac{1}{n}\right)\right)$$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n}\left(\tan ^{-1} n-\frac{\pi}{2}\right)$$

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum_{n=1}^{\infty}\left(1-\frac{1}{n}\right)^{n \cdot n}$$

Does \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{p}}\) converge if \(p\) is large enough? If so, for which \(p\) ?

Does \(\sum_{n=1}^{\infty}\left(\frac{(\ln n)}{n}\right)^{p}\) converge if \(p\) is large enough? If so, for which \(p\) ?

For which \(p\) does the series \(\sum_{n=1}^{\infty} 2^{p n} / 3^{n}\) converge?

For which \(p>0\) does the series \(\sum_{n=1}^{\infty} \frac{n^{p}}{2^{n}}\) converge?