Problem 6
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin (1 / n)}{n} $$
Problem 6
Use the root test to determine whether the series \(\sum_{n=1}^{\infty} 1 / n^{n}\) converges or diverges.
Problem 7
Compute the first four partial sums S1,…,S4 for the series having nth term an starting with n=1 as follows. $$ a_{n}=\sin (n \pi / 2) $$
Problem 7
For any real number \(p\), the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{p}} $$ is called a \(p\) -series.
Problem 7
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{1}{n !} $$
Problem 8
Find a formula \(a_{n}\) for the \(n\) th term of the geometric sequence whose first term is \(a_{1}=1\) such that \(\frac{a_{n+1}}{a_{n}}=10\) for \(n \geq 1\).
Problem 8
Compute the first four partial sums S1,…,S4 for the series having nth term an starting with n=1 as follows. $$ a_{n}=(-1)^{n} $$
Problem 8
For the series \(\sum_{n=1}^{\infty} \frac{2^{n}}{3^{n}+n}\), determine which convergence test is the best to use and explain why.
Problem 8
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin (1 / n)}{n} $$
Problem 8
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n+3}\)
