\text { (7) Suppose that } \lim _{n \rightarrow \infty} \sqrt{n} a_{n} \geq 1 \text { . Test the following series for convergence, } \sum_{n=1}^{\infty} \frac{a_{n}+\frac{1}{\sqrt{n}}}{n^{2} a_{n}^{2}}

1. Prove that convergence of {sn} implies convergence of {|Sn}. Is the converse true?

(4) Let {an} be a sequence of positive numbers. Define the sequence {bn} by b_{n}=\frac{\sin a_{n}+\sqrt{a_{n}^{2}+1}}{a_{n}+1} Show that {b„} has a convergent subsequence.

4. [5pt] Raytracing is an algorithm that involves finding the point at which a ray (a line with a direction and an origin) intersects a curve or surface. We will consider a ray intersecting with an ellipse. The general equation for an ellipse is \left(\frac{x}{\alpha}\right)^{2}+\left(\frac{y}{\beta}\right)^{2}-1=0 and the equation for a ray starting from the point Po = [xo, Yo] in the direction Vo = [u0, vo], is \mathbf{R}(t)=\left[x_{0}+t u_{0}, y_{0}+t v_{0}\right] where t e [0, x) parameterizes the ray. In this problem we will take a = 3, B = 2, Po = [0, b],Vo = [1, –0.3]. Using your favorite root finding algorithm write a code which computes theintersection of the given ray and the ellipse and plot your results. . (a) Plug the equation for the ray, R(t), into the equation for the ellipse and analytically (with pen and paper) solve for the value of t which gives the point of intersection, call it ti. li.(b) Perform the same calculation numerically using your favorite root finder. Report your answer to within an error of 10-6 and justify how you found the minimum number of iterations required to achieve this tolerance. Also report the point of intersection P1 = R(t1)

(10) Find the upper and lower limits of each sequence: \text { (a) }\left\{\frac{2-(-1)^{n} n}{3 n+2}\right\} \text { (b) }\left\{\frac{2 n-1}{n} \sin \frac{n \pi}{6}\right\} \text { . }

\text { (5) Suppose } \lim _{n \rightarrow \infty} \frac{u_{n}-1}{a_{n}+1}=0 . \text { Show that } \lim _{n \rightarrow \infty} a_{n}=1 \text { . }

Prove the following statements. [Each part is worth 5pt (a) Let {rn} be a converging sequence in a metric space X and let x € X be its limit. Use the definition of compactness to show that the set {x}U{xn} is compact. (b) Show that a subset of a metric space X is closed if and only if its intersection with every compact subset of X is closed.

(9) Let {an} be a bounded sequence in R such that every convergent subsequence of {an} has the same limit a. Prove that {an} converges to a.

The rate at which water flows through Table Rock Dam on the White Rive in Branson, MO, is measured in thousands of cubic feet per second (TCFS). As engineers open the floodgates, flow rates are recorded according to the following chart.' 1. What question might the engineers be interested in answering using this data? Why would it be easier to measure flow rate instead of directly what they want to measure? 2. What definite integral describes what the engineers are trying to measure about the dam in the 60-second time period? 3. Graph the data. 4. Think about the situation that is being measured, if you were to connect the data points would it be smooth curves or straight lines? Why? 5. Use the data to calculate Mn for the largest possible n to approximate what you are trying to measure in (2). Do you think Mn over- or under-estimates the exact value of the integral?Why? 6. Approximate the integral in (2) by calculating Sn for the largest possible n. 2000+2100+2400+3000+3900+5100+6500Compute S, and60What7quantity do both of these values estimate? Which is a moreaccurate approximation?

3Evaluate without using an electronic calculator. Take care to show how you obtained your answer: i) log, 4 ii) log, 4 iii) log, 3 iv) log, 2