Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

If \(\displaystyle \sum_{n=1}^\infty a_n=5\) what can be said about \(\displaystyle\sum_{n=1}^\infty b_n\text{?}\)

1. \(\displaystyle\sum_{n=1}^\infty b_n\) converges but the value cannot be determined.

2. \(\displaystyle\sum_{n=1}^\infty b_n\) converges to \(3\cdot 5=15\text{.}\)

3. \(\displaystyle\sum_{n=1}^\infty b_n\) converges to some value other than 15.

4. \(\displaystyle\sum_{n=1}^\infty b_n\) diverges.

5. It cannot be determined whether \(\displaystyle\sum_{n=1}^\infty b_n\) converges or diverges.

(b)

If \(\displaystyle \sum_{n=1}^\infty a_n\) diverges, what can be said about \(\displaystyle\sum_{n=1}^\infty b_n\text{?}\)

1. \(\displaystyle\sum_{n=1}^\infty b_n\) converges but the value cannot be determined.

2. \(\displaystyle\sum_{n=1}^\infty b_n\) converges and the value can be determined.

3. \(\displaystyle\sum_{n=1}^\infty b_n\) diverges.

4. It cannot be determined whether \(\displaystyle\sum_{n=1}^\infty b_n\) converges or diverges.

(a)

What can we say about the series

1. \(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) converges to \(3\cdot 2=6\text{.}\)

2. \(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) converges to some value other than 6.

3. \(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) diverges.

(b)

What do you think we can say about the series

1. \(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) converges to \(3\cdot 2=6\text{.}\)

2. \(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) converges to some value other than 6.

3. \(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) diverges.

(a)

What can we say about the series

1. \(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) converges to a known value we can compute.

2. \(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) converges to some unknown value.

3. \(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) diverges.

(b)

What do you think we can say about the series

1. \(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) converges to a known value we can compute.

2. \(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) converges to some unknown value.

3. \(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) diverges.

(a)

Let \(b_n=\frac{1}{n}\text{.}\) Compute \(\displaystyle \lim_{n\to\infty}\frac{\frac{1}{n}}{\frac{1}{2^n}}\text{.}\)

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

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

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

4. \(1\text{.}\)

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

(b)

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

(c)

Let \(\displaystyle b_n=\frac{1}{n^2}\text{.}\) Compute \(\displaystyle \lim_{n\to\infty}\frac{\frac{1}{n^2}}{\frac{1}{2^n}}\text{.}\)

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

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

3. \(1\text{.}\)

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

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

(d)

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

(e)

Let \(\displaystyle\sum a_n\) and \(\displaystyle\sum b_n\) be series with positive terms. If

diverges, can we conclude that \(\displaystyle \sum b_n\) converges or diverges?

(a)

Compute

(b)

Does the geometric series \(\displaystyle \sum_{n=1}^\infty \frac{1}{4^n}\) converge or diverge by Fact 8.4.2?

(c)

Does \(\displaystyle \sum_{n=1}^\infty \frac{1}{4^n-1}\) converge or diverge?

(a)

To which of the following should we compare \(\displaystyle \{a_n\}=\left\{\frac{2}{\sqrt{n+3}}\right\}\text{?}\)

1. \(\displaystyle \left\{\frac{1}{n}\right\}\text{.}\)

2. \(\displaystyle \left\{\frac{1}{\sqrt{n}}\right\}\text{.}\)

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

4. \(\displaystyle \left\{\frac{1}{2^n}\right\}\text{.}\)

(b)

Compute \(\displaystyle \lim_{n\to\infty} \frac{b_n}{a_n}\text{.}\)

(c)

Compute \(\displaystyle \lim_{n\to\infty} \frac{a_n}{b_n}\text{.}\)

(d)

What is true about \(\displaystyle \lim_{n\to\infty} \frac{b_n}{a_n}\) and \(\displaystyle \lim_{n\to\infty} \frac{a_n}{b_n}\text{?}\)

1. Their values are reciprocals.

2. Their values negative reciprocals.

3. They are both positive finite constants.

4. Only one value is a finite positive constant.

5. One value is \(0\) and the other value is infinite.

(e)

Does the series \(\displaystyle \sum_{n=2}^\infty \frac{1}{\sqrt{n}}\) converge or diverge?

(f)

Using your chosen sequence and Fact 8.6.5, does \(\displaystyle \sum_{n=2}^\infty \frac{2}{\sqrt{n+3}}\) converge or diverge?

(a)

To which of the following should we compare \(\displaystyle \{x_n\}=\left\{\frac{3}{n^2+8n+5} \right\}\text{?}\)

1. \(\displaystyle \left\{\frac{1}{n}\right\}\text{.}\)

2. \(\displaystyle \left\{\frac{1}{\sqrt{n}}\right\}\text{.}\)

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

4. \(\displaystyle \left\{\frac{1}{2^n}\right\}\text{.}\)

(b)

Using your chosen sequence and Fact 8.6.5, does \(\displaystyle \frac{3}{n^2+8n+5}\) converge or diverge?

(a)

Suppose that \(\displaystyle \sum_{n=0}^\infty a_n\) converges. What could be said about \(\{b_n\}\text{?}\)

1. \(\displaystyle \sum_{n=0}^\infty b_n\) converges.

2. \(\displaystyle \sum_{n=0}^\infty b_n\) diverges.

3. Whether or not \(\displaystyle \sum_{n=0}^\infty b_n\) converges or diverges cannot be determined with this information.

(b)

Suppose that \(\displaystyle \sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{1}{n+1}\) which diverges. Which of the following statements are true?

1. \(\displaystyle 0\leq \frac{1}{2n^2} \leq \frac{1}{n+1}\) for each \(n \geq 1\) and \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n^2}\) is a convergent \(p\)-series where \(p=2\text{.}\)

2. \(\displaystyle 0\leq \frac{1}{2n}\leq \frac{1}{n+1}\) for each \(n \geq 1\) and \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n}\) is a divergent \(p\)-series where \(p=1\text{.}\)

(c)

Suppose that \(\displaystyle \sum_{n=0}^\infty a_n\) was some series that diverges. What could be said about \(\{b_n\}\text{?}\)

1. \(\displaystyle \sum_{n=0}^\infty b_n\) converges.

2. \(\displaystyle \sum_{n=0}^\infty b_n\) diverges.

3. Whether or not \(\displaystyle \sum_{n=0}^\infty b_n\) converges or diverges cannot be determined with this information.

(d)

Suppose that \(\displaystyle \sum_{n=0}^\infty b_n\) diverges. What could be said about \(\{a_n\}\text{?}\)

1. \(\displaystyle \sum_{n=0}^\infty a_n\) converges.

2. \(\displaystyle \sum_{n=0}^\infty a_n\) diverges.

3. Whether or not \(\displaystyle \sum_{n=0}^\infty a_n\) converges or diverges cannot be determined with this information.

(e)

Suppose that \(\displaystyle \sum_{n=0}^\infty b_n=\sum_{n=0}^\infty \frac{1}{3^n}\) which converges. Which of the following statements are true?

1. \(\displaystyle 0\leq \frac{1}{3^n} \leq \frac{1}{2^n}\) for each \(n\) and \(\displaystyle \sum_{n=0}^\infty \frac{1}{2^n}\) is a convergent geometric series where \(\displaystyle |r|=\frac{1}{2} \lt 1\text{.}\)

2. \(\displaystyle 0\leq \frac{1}{3^n} \leq 1\) for each \(n\) and \(\displaystyle \sum_{n=0}^\infty 1\) diverges by the Divergence Test.

(f)

Suppose that \(\displaystyle \sum_{n=0}^\infty b_n\) was some series that converges. What could be said about \(\{a_n\}\text{?}\)

1. \(\displaystyle \sum_{n=0}^\infty a_n\) converges.

2. \(\displaystyle \sum_{n=0}^\infty a_n\) diverges.

3. Whether or not \(\displaystyle \sum_{n=0}^\infty a_n\) converges or diverges cannot be determined with this information.

(a)

How might we best utilize Fact 8.6.12 to determine the convergence of \(\displaystyle \sum_{n=0}^\infty a_n\) or \(\displaystyle \sum_{n=0}^\infty b_n\text{?}\)

1. Since \(a_n\) is sometimes greater than, and sometimes less than \(b_n\text{,}\) there is no way to utilize Fact 8.6.12.

2. Since initially, we have \(b_n\leq a_n\text{,}\) we can utilize Fact 8.6.12 by assuming \(a_n\geq b_n\text{.}\)

3. Since we can rewrite \(\displaystyle \sum_{n=0}^\infty a_n=\sum_{n=0}^{99} a_n+\sum_{n=100}^\infty a_n\) and \(\displaystyle \sum_{n=0}^\infty b_n=\sum_{n=0}^{99} b_n+\sum_{n=100}^\infty b_n\) and \(\displaystyle \sum_{n=0}^{99} a_n, \sum_{n=0}^{99} b_n\) are necessarily finite, we can compare \(\displaystyle \sum_{n=100}^\infty a_n, \sum_{n=100}^\infty b_n\) with Fact 8.6.12.

(a)

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

(b)

For which value \(k\) is \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k\text{?}\)

1. \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=0\text{.}\)

2. \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=1\text{.}\)

3. \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=2\text{.}\)

4. \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=3\text{.}\)

5. There is no \(k\) for which \(\displaystyle \frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k\text{.}\)

(c)

Use Fact 8.6.14 and compare \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n+3}\) to \(\displaystyle \sum_{n=1}^\infty \frac{1}{3n}\) to determine if \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n+3}\) converges or diverges.

(a)

Which series should we compare \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+5}\) to best utilize Fact 8.6.14?

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

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

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

4. \(\displaystyle\sum_{n=1}^\infty \frac{1}{n+5}\text{.}\)

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

6. \(\displaystyle\sum_{n=1}^\infty \frac{1}{2^n+5}\text{.}\)

(b)

Using your chosen series and Fact 8.6.14, does \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+5}\) converge or diverge?

(a)

(b)