### Question 3: Series Analysis Use the following information to answer questions 3a, 3b, 3c, and 3d: #### Given Series: \[ \sum_{n=1}^{\infty} \frac{n^2}{n^2 + 1} \] **3a) Test the series using the Divergence Test. What can you conclude?** **3b)** **Show that:** \[ f(x) = \frac{x^2}{x^2 + 1} \] Is decreasing on some interval that goes to infinity. **3c) Use the integral test to determine if the series is divergent or convergent.** **3d) Use a comparison test to determine if the series is divergent or convergent.** --- In this problem, the goal is to analyze the convergence or divergence of the given series using various tests, including the Divergence Test, the Integral Test, and a Comparison Test. Each test will provide different insights into the behavior of the series as \(n\) approaches infinity.

In her last semester at SPC, Polly Hedron needs to take Statistics, Composition 2, and Ethics.  Because Polly is registering early, she has 13 choices for her section of Statistics, 15 choices for her section of Composition, and 20 choices for her section of Ethics.  From how many possible schedules can Polly choose?

USA. Benjamin Banneker (1731-1806) is one of the great American heroes of all time. The son and grandson of Africans captured and enslaved, he was fortunate to have been born free. He inherited the family farm where he worked hard all of his life. As a child he had a few months of schooling. Everything else he had to do on his own. He loved mathematics and science but did not get a chance to learn astronomy until he was in his fifties. Then he performed a near miracle, learning it all in less than a year. In his hand-written journal, he recorded two right triangle examples that interested him. You can solve them here and remember the great man who enjoyed geometry. On the page in his journal next to the August 1775 astronomy calculations, Banneker wrote, in the style of that time: Suppose a ladder 60 feet long be placed in a Street so as to reach a window on the one Side 37 feet high, and without moving it at bottom, will reach another window on the other side of teh Street which is 23…

A student needs eight more classes to graduate. If she has met the prerequisites for all the classes, how many possible schedules for next semester could she make if she plans to take five classes? 

In 2004 it surfaced in Reader's Digest presented as having happened to a teacher at an unnamed community college. Four students walked in halfway through American history test my father was giving at the local community college. "Sorry," they said, "we had a flat tire." An understanding man, Dad said that if they could answer just one question correctly, he would give them each an "A" for the exam. The students agreed. So my father handed each one a piece of paper, placed them in four separate comers and said, "Write down which tire was flat." Kurt Smith What is the probability that these four students would give identical answers if they hadn't had a flat tire? Why?

Consider the following problem: "You and a friend go to dinner with four couples. Before dinner, everyone shakes hands with each person that they've never met. During dinner, you ask everyone how many hands they shook. You get the following answers: 0, 1, 2, 3, 4, 5, 6, 7, 8. How many hands did your friend shake?" Which of these problems below represent a simpler problem that could help you solve this one? O You and a friend go to dinner with two couples, and everyone shakes each othes person's hand. How many handshakes did your friend give? O You and a friend go to dinner with one couple, and everyone shakes hands with each person they've never met. During dinner, you ask everyone how many hands they shook. You get the following answers: 1, 1, 1. How many hands did you shake? O You go to dinner with two couples, and everyone shakes each other person's hand. How many handshakes total happened? O You and a friend go to dinner with one couple, and everyone shakes hands with each person…

Can you do this problem?

In this question, you are asked to comment on a student's attempt at answering the question detailed below. Read the question and the student's incorrect attempt first, then answer the questions below. The Question Ronan is building an extension to his house to facilitate a social space and games room. One of his passions is watching sport on a big screen, so he has purchased a wide-lens home cinema projector that he can connect to his laptop, instead of buying a large-screen TV. Having read the manufacturer's guide extensively, and researched his options on the internet for optimising picture quality, he decides that he would like to have a viewing area equivalent to having a 120-inch TV mounted on the wall. This means that the width of the viewing area will be 108 inches. The projector has a wide lens that covers an angle of 72°. He needs to position the projector centrally with respect to the viewing area, on a stand that you may assume is at the correct height for optimum picture…

I already have a,b,c and d. I just need help on E, F, G, and H

For the remainder of this question, we will consider one of the world's most popular video games, Minecraft (adapted from Mills & Mills, 2016). In this game, the character (you) can attempt to tame a wolf with a bone. Assume that you get 9 tries to tame the wolf, i.e. there are 9 bones. Each time you tame the wolf, the wolf becomes a dog and a new wolf appears which you can then attempt to also tame. This continues until you have used up all 9 bones, and the goal is to end up with as many dogs as possible at the end of the game. The probability of taming the wolf with each attempt is fixed at 0.54. Let X denote the number of dogs at the end of the game so that X BIN (9,0.54). Assume you are about to play the game. 12. What is the probability you will tame less than 4 dogs?