# CLEAR Calculus

## Lab 9: Series - part 2

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, $T_1 = 1 \text{ and } T_4 = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}.$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the location of the exact value.

2. Clearly mark the value of the sixth approximation on your graph and vertical number line. Represent the error, error bound, and range of possible values for this approximation.

3. Draw a zoomed-in sketch of the graph placing the value of $$T$$ at a reasonable location. Use your error bound to illustrate a region around $$T$$ within which approximations would be accurate to eight decimal places. Determine how many terms would be required to make the approximation accurate to eight decimal places. Clearly indicate the value for the $$n$$ that corresponds to this required approximation on your zoomed-in sketch.

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, $G_1 = 1 \text{ and } G_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}.$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the location of the exact value.

2. Clearly mark the value of the sixth approximation on your graph and vertical number line. Represent the error, error bound, and range of possible values for this approximation.

3. Draw a zoomed-in sketch of the graph placing the value of $$G$$ at a reasonable location. Illustrate a region around $$G$$ within which approximations would be accurate to eight decimal places. Determine how many terms would be required to make the approximation accurate to eight decimal places. Clearly indicate the value for the $$n$$ that corresponds to this required approximation on your zoomed-in sketch.

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, $R_1 = 1 \text{ and } R_4 = 1- \frac{1}{3} + \frac{1}{5} - \frac{1}{7}.$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the location of the exact value.

2. Clearly mark the value of the sixth approximation on your graph and vertical number line. Represent the error, error bound, and range of possible values for this approximation.

3. Draw a zoomed-in sketch of the graph placing the value of $$R$$ at a reasonable location. Illustrate a region around $$R$$ within which approximations would be accurate to eight decimal places. Determine how many terms would be required to make the approximation accurate to eight decimal places. Clearly indicate the value for the $$n$$ that corresponds to this required approximation on your zoomed-in sketch.

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, $D_1 = \frac{1}{2} \text{ and } D_4 = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8}.$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the location of the exact value.

2. Use the integral test to show that the series diverges.

3. Find the smallest value of $$n$$ (that you can) such that $$D_n > 10$$.

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, $M_1 = 1 \text{ and } M_5 = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!}.$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the locatio of the exact value.

2. Clearly mark the value of the sixth approximation on your graph and vertical number line. Represent the error, error bound, and range of possible values for this approximation.

3. Draw a zoomed-in sketch of the graph placing the value of $$M$$ at a reasonable location. Illustrate a region around $$M$$ within which approximations would be accurate to eight decimal places. Determine how many terms would be required to make the approximation accurate to eight decimal places. Clearly indicate the value for the $$n$$ that corresponds to this required approximation on your zoomed-in sketch.

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 ${\small P_4 = \frac{2}{9} + \frac{1}{72} \left[1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 \right] +\frac{2}{81} \left[1 + \frac{13}{18} + \left(\frac{13}{18}\right)^2 \right] + \frac{25}{648} \left[1 + \frac{25}{36} + \left(\frac{25}{36}\right)^2\right].}$

Complete Questions 1-3 for all three mystery constants in your new group. The instructions may be slightly different for each mystery constant.

1. Sketch a full-page graph and vertical number line showing the first 10 partial sum approximations and a representation for the location of the exact value.

2. Clearly mark the value of the sixth approximation on your graph and vertical number line. Represent the error for this approximation.

3. Find the first value of $$n$$ so that $$P_n$$ approximates $$P$$ accurate to eight decimal places.

Hint: Use geometric series to find a formula for the exact value of the error for $$P_n$$. Enter this in your calculator and use the table feature, scrolling to find the first time the error is less than $$5\times 10^{-9}$$.