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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}}
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}
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?
\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
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) }
Question 3. Assume La(x) is the line that is tangent to the graph of y = f(x) = x^2 at (a, f(a)). Let Aa be the point at which La(x) intersects the line x=0. (1) (1 point) Find the equation for La(x) for each a €R. (2) (1 point) Compute the distance between the points Ax and (x,f(x)). (3) (1 point) Find the rate of the change of the distance between A, and (x, f(x)) with respect to x. (4) (2 points) Use the definition of derivative to show that this rate is not defined at x= 0.
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}