84. Gluten is a mix of proteins found in soy and corn. seeds and nuts. wheat, rye, and barley. sugar, beans, and dairy.

1. Find a closed form for the following sequences: (a) The sequence starting at ao= 1/2, with recurrence relation a_{n}=\frac{1-a_{n-1}}{1+a_{n-1}} (b) The sequence starting at ao = –1, with recurrence relation (c) The sequence for arbitrary starting value ao. a_{n+1}=\frac{a_{n}}{1+a_{n}} a_{n+1}=1+3 a_{n}

1. (9 points) Determine whether the sequences listed below are increasing, decreasing,or not monotonic. \text { a. }\left\{\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{n !}\right\} \text { b. }\left\{\frac{(-1)^{n} n^{3}}{2 n^{3}+2 n^{2}+1}\right\} \text { c. }\left\{n^{2} e^{-n}\right\}

20.2.9 Prove that \frac{h}{2 \pi i} \int_{-\infty}^{\infty} \frac{e^{-i \omega t} d \omega}{E_{0}-i \Gamma / 2-\hbar \omega}=\left\{\begin{array}{ll} \exp \left(-\frac{\Gamma t}{2 \hbar}\right) \exp \left(-i \frac{E_{0} t}{\hbar}\right), & t>0 \\ 0, & t<0 \end{array}\right. This Fourier integral appears in a variety of problems in quantum mechanics: barrier penetration, scattering, time-dependent perturbation theory, and so on.

\text { Define the sequence }\left(a_{n}\right)_{n \geq 1} \text { recursively by } a_{1}=1 \text { and } a_{n+1}=\sqrt{2+a_{n}} \text { for all } n \geq 1 \text { i. Prove that for all } n \geq 1 \text { we have } 1 \leq a_{n} \leq 2 \text { ii. Prove that }\left(a_{n}\right)_{n \geq 1} \text { is a monotonic sequence. } iii. Deduce that the sequence converges with limit 2. iv. Show that for all n > 1 we have 2-a_{n} \leq \frac{1}{(2+\sqrt{3})^{n-1}}

Question 2. (4 points) Intensive harvesting of a population of a fish species can cause population extinction. We consider the following two models for harvesting of a given fish species and analyze how the extinction depends on the nature of the harvesting. The population size P (measured in thousands) is a function of harvesting effort h. (The harvesting effort is a mathematical measure of "fishing effort", which you are not expected to know in details.) \begin{aligned} &\text { Model }\\ &\text { 1: } \quad P(h)=\left\{\begin{array}{ll} 3(1-h) & \text { if } 0 \leq h \leq 1 \\ 0 & \text { if } h>1 \end{array}\right. \end{aligned} \begin{aligned} &\text { Model }\\ &\text { 2: } \quad P(h)=\left\{\begin{array}{ll} 1+\sqrt{4-3 h} & \text { if } 0 \leq h \leq \frac{4}{3} \\ 0 & \text { if } h>\frac{4}{3} \end{array}\right. \end{aligned} (1) (1 point) What is the initial population of this species when no harvesting efforts were applied at all? (2) (1 point) Draw the graph of each model. (3) (2 points) Here you will analyze each model in terms of the Intermediate Value Theorem by answering the following questions. Which model has a situation where a small change in harvesting effort causes a sudden extinction? In Model 2, is there a harvesting effort to obtain the population of 500?

3. [3 marks] Determine whether each of the following functions from Z to Z is one-to-one. If it is not, then justify. \begin{array}{l} \text { (a) } f(n)=n-10 \\ \text { (b) } f(n)=n^{2}+3 \\ \text { (c) } f(n)=16\left[\frac{n}{7}\right\rfloor \end{array}

Instructions 1. Create your own example of an alternating series that is (conditionally) convergent, but not absolutely convergent. Your series should be different than any of those in the notes or text examples. 2. Post your infinite series on Discussion Board on Canvas. Give a brief explanation of how you created your series. There are several different approaches you might take. 3. Peer response: look at the post from at least one other classmate and critique the method used to create the series. Do you think the method should work? Alternatively, if you believe your classmate's method works, describe another way the series might have been created.

\text { In a geometric sequence } u_{1}=125 \text { and } u_{6}=\frac{1}{25} (a) Find the value of r (the common ratio) (b) Find the largest value of n for which S. <156.22 (c) Explain why there is no value of n for which S, >160

Denote the Fourier series of f(x)=\left\{\begin{array}{ll} -x, & -\pi \leq x<0 \\ x, & 0 \leq x<\pi \end{array}\right. by F(x), where F(x) has a period 27. Show that F(x)=\frac{\pi}{2}-\frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos (2 m-1) x}{(2 m-1)^{2}} \text { For }-2 \pi<x<2 \pi \text { siketch the graph of } F(x) Deduce that 1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\cdots=\frac{\pi^{2}}{8}

Find the sum of the solutions to the equation given. 2\left(9^{x}\right)-3^{x+2}+54=4\left(3^{x+1}\right) \text { (note, this is similar to problem } 8 \text { from the practice, so show all steps in order to get credit) }