# CLEAR Calculus

## Lab 8: Series - part 1

Mystery Constant T

A chemical process causes temperature fluctuations inside a reactor, described by the equation for $$T$$ given below (in units of 100$$^\circ$$ C). $T= \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \cdots.$ (Note that $$n! = 1\cdot 2\cdot 3\cdots n$$ and by convention, $$0!=1$$.)

Mystery Constant G

In an experiment, the mass of an incubated yeast cell colony was determined to double every 43 hours. Consequently the lab was able to compute that after 43 hours, the instantaneous rate of growth (in grams per hour) would be $$G$$ times the initial mass, where $$G$$ is given by the series $G = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots.$

Mystery Constant R

The proportion of the area of a square covered by an inscribed circle is some number $$R$$ (between 0 and 1) given by the infinite sum $R = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots.$

Mystery Constant D

A book can be placed on the edge of a table so that $$\frac{1}{2}$$ of it is on the table and $$\frac{1}{2}$$ extends off of the table. If the book is placed any farther it will fall off. A second book can then be placed underneath the first one so that $$\frac{1}{4}$$ of its length extends beyond table, so the stack of two books extends $$\frac{3}{4}$$ a book length off of the table. The farthest such a stack could theoretically extend, $$D$$, is therefore given by the infinite sum $D = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{12} + \frac{1}{14} + \cdots.$

Mystery Constant M

An investment with a 5% interest rate compounded continuously over 20 years will earn an amount M times the original investment in interest, where $$M$$ is the infinite sum $M = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + \cdots.$

Mystery Constant P

The probability of winning a round in the dice game "craps" is a number $$P$$ between 0 and 1, given by the infinite sum \begin{align*} P =& \frac{2}{9} + \frac{1}{72} \left[1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3 + \left(\frac{3}{4}\right)^4 + \cdots\right]\\ & \quad + \frac{2}{81} \left[1 + \frac{13}{18} + \left(\frac{13}{18}\right)^2 + \left(\frac{13}{18}\right)^3 + \left(\frac{13}{18}\right)^4 + \cdots\right]\\ & \quad + \frac{25}{648} \left[1 + \frac{25}{36} + \left(\frac{25}{36}\right)^2 + \left(\frac{25}{36}\right)^3 + \left(\frac{25}{36}\right)^4 + \cdots\right] \end{align*}

Lab 8: Mystery Constant T

A chemical process causes temperature fluctuations inside a reactor, described by the equation for $$T$$ given below (in units of 100$$^\circ$$ C). $T= \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \cdots.$ (Note that $$n! = 1\cdot 2\cdot 3\cdots n$$ and by convention, $$0!=1$$.)

You can approximate $$T$$ by computing a partial sum $$T_n$$. For example, $T_1 = 1 \text{ and } T_4 = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}.$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$T_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$T$$ on a number line and as the graph of a sequence.

Choose an appropriate scale and make the diagrams large - you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. item Even though you don't know what it is, place a mark to represent the exact value of $$T$$ at a reasonable place on both the vertical number line and the $$T_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Use this bound to find a range of possible values for $$T$$. Represent both the bound on the error and the range of possible values in your diagram.

4. Find an approximation that is accurate to two decimal places (error $$<$$0.005). Explain how you know your approximation satisfies this requirement.

Lab 8: Mystery Constant G

In an experiment, the mass of an incubated yeast cell colony was determined to double every 43 hours. Consequently the lab was able to compute that after 43 hours, the instantaneous rate of growth (in grams per hour) would be $$G$$ times the initial mass, where $$G$$ is given by the series $G = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots.$ You can approximate $$G$$ by computing a partial sum $$G_n$$. For example, $G_1 = 1 \text{ and } G_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}.$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$G_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$G$$ on a number line and as the graph of a sequence.

Choose an appropriate scale and make the diagrams large -- you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. Even though you don't know what it is, place a mark to represent the exact value of $$G$$ at a reasonable place on both the vertical number line and the $$G_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Use this bound to find a range of possible values for $$G$$. Represent both the bound on the error and the range of possible values in your diagram.

4. Find an approximation that is accurate to two decimal places (error $$<$$ 0.005). Explain how you know your approximation satisfies this requirement.

Lab 8: Mystery Constant R

The proportion of the area of a square covered by an inscribed circle is some number $$R$$ (between 0 and 1) given by the infinite sum $R = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots.$

You can approximate $$R$$ by computing a partial sum $$R_n$$. For example, $R_1 = 1 \text{ and } R_4 = 1- \frac{1}{3} + \frac{1}{5} - \frac{1}{7}.$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$R_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$R$$ on a number line and as the graph of a sequence.

Choose an appropriate scale and make the diagrams large -- you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. Even though you don't know what it is, place a mark to represent the exact value of $$R$$ at a reasonable place on both the vertical number line and the $$R_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Use this bound to find a range of possible values for $$R$$. Represent both the bound on the error and the range of possible values in your diagram.

4. Find an approximation that is accurate to two decimal places (error $$<$$ 0.005). Explain how you know your approximation satisfies this requirement.

Lab 8: Mystery Constant D

A book can be placed on the edge of a table so that $$\frac{1}{2}$$ of it is on the table and $$\frac{1}{2}$$ extends off of the table. If the book is placed any farther it will fall off. A second book can then be placed underneath the first one so that $$\frac{1}{4}$$ of its length extends beyond table, so the stack of two books extends $$\frac{3}{4}$$ a book length off of the table. The farthest such a stack could theoretically extend, $$D$$, is therefore given by the infinite sum $D = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{12} + \frac{1}{14} + \cdots.$ You can approximate $$D$$ by computing a partial sum $$D_n$$. For example, $D_1 = \frac{1}{2} \text{ and } D_4 = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8}.$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$D_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$D$$ on a number line a as the graph of a sequence.

Choose an appropriate scale and make the diagrams large -- you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. Even though you don't know what it is, place a mark to represent the exact value of $$D$$ at a reasonable place on both the vertical number line and the $$D_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Ask for a hint if you spend more than 15 minutes on this.

4. Determine the smallest value of $$n$$ so that $$D_n > 1$$. Then find the smallest values of $$n$$ so that $$D_n > 2$$ and $$D_n > 3$$. What does this pattern suggest. Explain what implications this has for your previous answers to Questions 1-4?

Lab 8: Mystery Constant M

An investment with a 5% interest rate compounded continuously over 20 years will earn an amount M times the original investment in interest, where $$M$$ is the infinite sum $M = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + \cdots.$

You can approximate $$M$$ by computing a partial sum $$M_n$$. For example, $M_1 = 1 \text{ and } M_5 = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!}.$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$M_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$M$$ on a number line and as the graph of a sequence.

Choose an appropriate scale and make the diagrams large -- you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. Even though you don't know what it is, place a mark to represent the exact value of $$M$$ at a reasonable place on both the vertical number line and the $$M_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Use this bound to find a range of possible values for $$M$$. Represent both the bound on the error and the range of possible values in your diagram. Ask for a hint if you spend more than 15 minutes on this.

4. Find an approximation that is accurate to two decimal places (error $$<$$0.005). Explain how you know your approximation satisfies this requirement.

Lab 8: Mystery Constant P

The probability of winning a round in the dice game "craps" is a number $$P$$ between 0 and 1, given by the infinite sum \begin{align*} P =& \frac{2}{9} + \frac{1}{72} \left[1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3 + \left(\frac{3}{4}\right)^4 + \cdots\right]\\ & \quad + \frac{2}{81} \left[1 + \frac{13}{18} + \left(\frac{13}{18}\right)^2 + \left(\frac{13}{18}\right)^3 + \left(\frac{13}{18}\right)^4 + \cdots\right]\\ & \quad + \frac{25}{648} \left[1 + \frac{25}{36} + \left(\frac{25}{36}\right)^2 + \left(\frac{25}{36}\right)^3 + \left(\frac{25}{36}\right)^4 + \cdots\right] \end{align*}

You can approximate $$P$$ by computing a partial sum $$P_n$$. For example, $P_1 = \frac{2}{9}$  and ${\small P_4 = \frac{2}{9} + \frac{1}{72} \left[1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 \right] +\frac{2}{81} \left[1 + \frac{13}{18} + \left(\frac{13}{18}\right)^2 \right] + \frac{25}{648} \left[1 + \frac{25}{36} + \left(\frac{25}{36}\right)^2\right].}$

Pre-Lab Assignment: Answer the following questions individually and bring your write-up to class.

1. Make a table of values for $$n$$ and $$P_n$$ for $$n = 1$$ to $$n = 10$$.

2. Plot the first 10 approximations for $$P$$ on a number line and as the graph of a sequence.

Choose an appropriate scale and make the diagrams large -- you will be adding several things to it.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

1. Even though you don't know what it is, place a mark to represent the exact value of $$P$$ at a reasonable place on both the vertical number line and the $$P_n$$-axis of the graph.

2. Clearly mark the value of the fifth approximation on your graph and number line. Represent the error for this approximation on both diagrams, and write an algebraic expression for the error.

3. Determine a bound on the error for your fifth approximation. Ask for a hint if you spend more than 15 minutes on this.

4. Find the smallest value of $$n$$ so that $$P_n$$ is accurate to two decimal places (error $$<$$0.005). Explain how you know your approximation satisfies this requirement.

Mystery Constants TG, and R - HINT

Note that these are alternating series where the magnitude of the terms decrease monotonically to zero. This means that the series converge by the alternating series test. Additionally the partial sums will alternate between being underestimates and overestimates for the limits $$T$$, $$G$$, and $$R$$. So an error bound is simply the next term in the sequence (the difference between an underestimate and overestimate).

Mystery Constant D - HINT (and apology)

This is a bit of a trick question (Sorry). This series diverges (goes to infinity)! To see this is the case group the terms as follows:  \begin{align*} D & = \frac{1}{2} +\left(\frac{1}{4}\right) \left(\frac{1}{6} + \frac{1}{6}\right) + \left(\frac{1}{10}+ \frac{1}{12} + \frac{1}{14} + \frac{1}{16}\right) \\ & \qquad + \left(\frac{1}{18} + \frac{1}{20} + \frac{1}{22} + \frac{1}{24} + \frac{1}{26} + \frac{1}{28} + \frac{1}{30} + \frac{1}{32}\right) + \left(\frac{1}{34}\right.\end{align*} doubling the number of terms in successive sets of parentheses. Then  \begin{align*} D > & \frac{1}{4} + \left(\frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16}+ \frac{1}{16} + \frac{1}{16} + \frac{1}{16}\right)\\ & \qquad + \left(\frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32} + \frac{1}{32}\right) + \left(\frac{1}{64}\right.\end{align*}

But each grouping is equal to $$\frac{1}{4}$$. So if you go out far enough, you can always add more than $$\frac{1}{4}$$ and continue doing so.

Replace Questions 1-3 with this: If you could harness the combined computing power of the entire world (in 2012) running for the entire existence of the universe, you would reach a partial sum in the neighborhood of $D_{848,929,773,798,124,602,508,711,960,865,872,012} = 41.999824\dots$ Suppose someone suggested that this probably means $$D = 42$$. Pick an error bound of 0.5 and explain to your friend why even though $$D_{848,929,773,798,124,602,508,711,960,865,872,012}$$ is certainly within that error bound of 42, there will eventually be a $$D_n$$ that is outside of this range.

Illustrate your reasoning on a full-page graph. Then focus on clearly answering Question 4.

Mystery Constants M - HINT

Suppose you stop at $$M_4$$ . Then the remaining terms are $\text{error } = \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + \frac{1}{8!} +\frac{1}{9!} +\cdots$ so $\text{error } < \frac{1}{5!} + \frac{1}{5!5} + \frac{1}{5!5^2} + \frac{1}{5!5^3} +\frac{1}{5!5^4} +\frac{1}{5!5^5} + \cdots$ That is, you can bound the error with a geometric series.

Mystery Constants P - HINT

Your sum is a geometric series. So you can compute the exact value of $$P$$ using techniques from that section. You can also use geometric series to find a closed form formula for $$P_n$$ or for the error for $$P_n$$.