Problem 4
Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{1}=1, a_{2}=1 \text { and } a_{n+2}=a_{n}+a_{n+1} \text { for } n \geq 1 $$
Problem 5
Compute the first four partial sums S1,…,S4 for the series having nth term an starting with n=1 as follows. $$ a_{n}=n $$
Problem 5
Absolute versus Conditional Convergence For each of the following series, determine whether the series converges absolutely, converges conditionally, or diverges. a. \(\sum_{n=1}^{\infty}(-1)^{n+1} /(3 n+1)\) b. \(\sum_{n=1}^{\infty} \cos (n) / n^{2}\)
Problem 5
Using the Integral Test For each of the following series, use the integral test to determine whether the series converges or diverges a. \(\sum_{n=1}^{\infty} 1 / n^{3}\)
Problem 5
For each of the following series, use the root test to determine whether the series converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{\left(n^{2}+3 n\right)^{n}}{\left(4 n^{2}+5\right)^{n}}\) b. \(\sum_{n=1}^{\infty} \frac{n^{n}}{(\ln (n))^{n}}\)
Problem 5
Use the comparison test to determine whether the following series converge. $$ \sum_{n=2}^{\infty} \frac{1}{(n \ln n)^{2}} $$
Problem 5
Find an explicit formula for \(a_{n}\) where \(a_{1}=1\) and \(a_{n}=a_{n-1}+n\) for \(n \geq 2\).
Problem 6
Determine whether the series \(\sum_{n=1}^{\infty}(-1)^{n+1} n /\left(2 n^{3}+1\right)\) converges absolutely, converges conditionally, or diverges.
Problem 6
Use the integral test to determine whether the series \(\sum_{n=1}^{\infty} \frac{n}{3 n^{2}+1}\) converges or diverges.
Problem 6
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin (1 / n)}{n} $$
