Problem 2
Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{n}=n^{2}-1 \text { for } n \geq 1 $$
Problem 2
For each of the following series, use the ratio test to determine whether the series converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{2^{n^{n}}}{n !}\) b. \(\sum_{n=1}^{\infty} \frac{n^{n}}{n !} \sum_{n=1}^{\infty} \frac{(-1)^{n}(n !)^{2}}{(2 n) !}\) c. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(n !)^{2}}{(2 n) !}\)
Problem 3
Use the ratio test to determine whether the series \(\sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}}\) converges or diverges.
Problem 3
Using sigma notation, write the following expressions as infinite series. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots $$
Problem 3
Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{1}=1 \text { and } a_{n}=a_{n-1}+n \text { for } n \geq 2 $$
Problem 3
Estimating the Remainder of an Alternating Series Consider the alternating series $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} $$ Use the remainder estimate to determine a bound on the error \(R_{10}\) if we approximate the sum of the series by the partial sum \(S_{10}\)
Problem 3
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{1}{2(n+1)} $$
Problem 3
What does the divergence test tell us about the series \(\sum_{n=1}^{\infty} \cos \left(1 / n^{2}\right) ?\)
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 4
Using sigma notation, write the following expressions as infinite series. $$ \sin 1+\sin 1 / 2+\sin 1 / 3+\sin 1 / 4+\cdots $$
