4. Determine whether the series

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}

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

\text { 4. Let } C([-3,3]) \text { be the vector space of continuous functions } f:[-3,3] \rightarrow \mathbb{R} \text { with the norm of uniform convergence }\|f\|_{\infty}:=\max _{x \in[-3,3]}|f(x)| \text {. } \text { (i) Consider the linear mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text {, } L f:=\int_{-3}^{3} x f(x) d x \text { Prove that the mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text { is continuous. } \text { (ii) Consider the mapping } F: C([-3,3]) \rightarrow \mathbb{R} \text {, } F(f):=\int_{-3}^{3}|x \| f(x)| d x Why you can not use the same strategy as in part (i) to prove that F is[2 Marks]continuous?

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) }

OPTIONAL Let (an) be a monotone decreasing zero sequence. Show that the sequence b_{n}:=a_{1}-a_{2}+a_{3}-a_{4} \pm \cdots \pm a_{n}=\sum_{k=1}^{n}(-1)^{n+1} a_{n} is convergent.

Find and simplify a formula for a,, the term of the given sequence. (Hint: Decide what kind of sequence it is then use information from within the section.) 4, 20, 100, 500, 2500.... The simplified formula for the term is a,- 8 X D.O G Nokia c E