Problem 12
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n !}\)
Problem 12
Find a formula for the general term \(a_{n}\) of each of the following sequences. $$ \\{1,0,-1,0,1,0,-1,0, \ldots\\} \text { (Hint: Find where } \sin x \text { takes these values) } $$
Problem 12
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{n} $$
Problem 13
Find a formula for the general term \(a_{n}\) of each of the following sequences. $$ \\{1,-1 / 3,1 / 5,-1 / 7, \ldots\\} $$
Problem 13
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sqrt[4]{n}}{\sqrt[3]{n^{4}+n^{2}}} $$
Problem 13
For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n}{n+2} $$
Problem 13
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3^{n}}{n !}\)
Problem 13
For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{n}{5 n^{2}-3}\)
Problem 14
Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty}\left(\frac{\ln n}{n}\right)^{2} $$
Problem 14
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n-1}{n}\right)^{n}\)
