Problem 8
Testing for Convergence of \(p\) -series For each of the following series, determine whether it converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\) b. \(\sum_{n=1}^{\infty} \frac{1}{n^{2 / 3}}\)
Problem 9
In the following exercises, compute the general term an of the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. $$ S_{n}=1-\frac{1}{n}, n \geq 2 $$
Problem 9
Find a formula \(a_{n}\) for the \(n\) th term of the geometric sequence whose first term is \(a_{1}=3\) such that \(\frac{a_{n+1}}{a_{n}}=1 / 10\) for \(n \geq 1\).
Problem 9
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sqrt{n}+1}{\sqrt{n}+3}\)
Problem 9
Does the series \(\sum_{n=1}^{\infty} \frac{1}{n^{5 / 4}}\) converge or diverge?
Problem 9
Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ a_{n}=1 / n ! $$
Problem 9
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin ^{2} n}{n^{2}} $$
Problem 10
State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt{n+3}}\)
Problem 10
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin (1 / n)}{\sqrt{n}} $$
Problem 10
In the following exercises, compute the general term an of the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. $$ S_{n}=\frac{n(n+1)}{2}, n \geq 1 $$
