Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Which of these series most closely resembles \(\displaystyle \sum_{n=0}^\infty \frac{2^n}{3^n-2}\text{?}\)

1. \(\displaystyle \sum_{n=0}^\infty \frac{2}{3}\text{.}\)

2. \(\displaystyle \sum_{n=0}^\infty \frac{2}{3}n\text{.}\)

3. \(\displaystyle \sum_{n=0}^\infty \left(\frac{2}{3}\right)^n\text{.}\)

(b)

Based on your previous choice, do we think this series is more likely to converge or diverge?

(c)

Find \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=\lim_{n\to\infty}\frac{2^{n+1}(3^n-2)}{(3^{n+1}-2)2^n}=\lim_{n\to\infty}\frac{2\cdot 2^{n}(3^n-2)}{3(3^{n}-\frac{2}{3})2^n}.\)

1. \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=0\text{.}\)

2. \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=\frac{2}{3}\text{.}\)

3. \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=1\text{.}\)

4. \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=2\text{.}\)

5. \(\displaystyle \lim_{n\to\infty} \frac{\frac{2^{n+1}}{3^{n+1}-2}}{\frac{2^n}{3^n-2}}=3\text{.}\)

(a)

Does \(\displaystyle \sum_{n=0}^\infty a_n=\sum_{n=0}^\infty \frac{3}{2^n}\) converge?

(b)

Find \(\displaystyle\frac{a_{n+1}}{a_n}\text{.}\)

1. \(2\text{.}\)

2. \(\displaystyle \frac{1}{2}\text{.}\)

3. \(\displaystyle \frac{2^n}{2^n+1}\text{.}\)

4. \(\displaystyle \frac{9}{2^{2n+1}}\text{.}\)

5. \(\displaystyle \frac{9}{2^{n+2}}\text{.}\)

(c)

Find \(\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|.\)

1. \(-\infty\text{.}\)

2. \(0\text{.}\)

3. \(\displaystyle \frac{1}{2}\text{.}\)

4. \(2\text{.}\)

5. \(\infty\text{.}\)

(a)

Does \(\displaystyle \sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{n^2}{n+1}\) converge?

(b)

Find \(\displaystyle\frac{a_{n+1}}{a_n}\text{.}\)

1. \(\displaystyle \frac{n+1}{2}\text{.}\)

2. \(\displaystyle \frac{(n^2+1)(n+1)}{(n+2)n^2}\text{.}\)

3. \(\displaystyle \frac{(n+1)^2}{n+2}\text{.}\)

4. \(\displaystyle \frac{1}{2}\text{.}\)

5. \(\displaystyle \frac{(n+1)n^2}{n+2}\text{.}\)

(c)

Find \(\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|.\)

1. \(-\infty\text{.}\)

2. \(0\text{.}\)

3. \(\displaystyle \frac{1}{2}\text{.}\)

4. \(2\text{.}\)

5. \(\infty\text{.}\)

(a)

Does \(\displaystyle \sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{1}{n}\) converge?

(b)

Find \(\displaystyle\frac{a_{n+1}}{a_n}\text{.}\)

(c)

Find \(\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|.\)

(a)

Does \(\displaystyle \sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{(-1)^n}{n}\) converge?

(b)

Find \(\frac{a_{n+1}}{a_n}\text{.}\)

(c)

Find \(\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|.\)

(a)

Which of the following is \(a_n\text{?}\)

1. \(n^2\text{.}\)

2. \(n!\text{.}\)

3. \(\displaystyle\frac{n^2}{n!}\text{.}\)

(b)

Which of the following is \(a_{n+1}\text{?}\)

1. \(\displaystyle\frac{n^2}{n!}\text{.}\)

2. \(\displaystyle(n+1)^2\text{.}\)

3. \(\displaystyle(n+1)!\text{.}\)

4. \(\displaystyle\frac{(n+1)^2}{(n+1)!}\text{.}\)

5. \(\displaystyle\frac{n^2+1}{n!+1}\text{.}\)

(c)

Which of the following is \(\displaystyle\left|\frac{a_{n+1}}{a_n}\right|\text{?}\)

1. \(\displaystyle\frac{(n+1)^2n^2}{(n+1)!n!}\text{.}\)

2. \(\displaystyle\frac{(n+1)^2n!}{(n+1)!n^2}\text{.}\)

3. \(\displaystyle\frac{(n+1)!n!}{(n+1)^2n^2}\text{.}\)

4. \(\displaystyle\frac{(n+1)!n^2}{(n+1)^2n!}\text{.}\)

(d)

Using the fact \((n+1)!=(n+1)\cdot n!\text{,}\) simplify \(\displaystyle\left|\frac{a_{n+1}}{a_n}\right|\) as much as possible.

(e)

Find \(\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\text{.}\)

(f)

Does \(\displaystyle\sum_{n=0}^\infty \frac{n^2}{n!}\) converge?

(a)

What is \(a_n\text{?}\)

(b)

Which of the following is \(\displaystyle\sqrt[n]{|a_n|}\text{?}\)

1. \(\displaystyle \frac{n+1}{9}\text{.}\)

2. \(\displaystyle \frac{n}{9}\text{.}\)

3. \(n\text{.}\)

4. \(9\text{.}\)

5. \(\displaystyle \frac{1}{9}\text{.}\)

(c)

Find \(\displaystyle\lim_{n\rightarrow\infty}\sqrt[n]{|a_n|}\text{.}\)

(d)

Does \(\displaystyle \sum_{n=1}^\infty \frac{n^n}{9^n}\) converge?

(a)

\(\displaystyle \sum_{n=1}^\infty \displaystyle\left(\frac{1}{1+n}\right)^n\)

(b)

\(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{2^n}{n^n}\)

(c)

\(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{(2n)!}{(n!)(n!)}\)

(d)

\(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{4^n(n!)(n!)}{(2n)!}\)

(a)

Use the root test to check for convergence of this series.

(b)

Use the ratio test to check for convergence of this series.

(c)

Use the comparison (or limit comparison) test to check for convergence of this series.

(d)

Find the sum of this series.

(a)

Let \(\displaystyle \alpha=\lim_{n\to\infty}\ln(n^{1/n})=\lim_{n\to\infty}\frac{1}{n}\ln(n)\text{.}\) Find \(\alpha\text{.}\)

(b)

Recall that \(\displaystyle \lim_{n\to\infty}n^{1/n}=\lim_{n\to\infty} e^{\ln(n^{1/n})}=e^\alpha.\) Find \(\displaystyle \lim_{n\to\infty}n^{1/n}\text{.}\)

(c)

Find \(\displaystyle \lim_{n\to\infty} \sqrt[n]{\frac{n}{3^n}}=\lim_{n\to\infty}\left(\frac{n}{3^n}\right)^{1/n}=\lim_{n\to\infty}\frac{n^{1/n}}{(3^n)^{1/n}}\text{.}\)

(d)

Does \(\displaystyle\sum_{n=1}^\infty \frac{n}{3^n}\) converge?

(a)

Use the root test to check for convergence of this series.

(b)

Use the ratio test to check for convergence of this series.

(c)

Use the comparison (or limit comparison) test to check for convergence of this series.