AP Calculus BC Review Sequences and Convergence

One important topic that shows up on the AP Calculus BC exam (but not on the AB) is sequences. This review article is dedicated to sequences and their convergence properties. What are Sequences? Basically, a sequence is just a list of numbers. However, its also much more than that. We usually study infinite sequences, those that go on forever according to some rule or pattern. Furthermore, we are often interested in determining whether a sequence converges (that is, approaches some fixed value) or not. Definition and Notation A sequence is a list of (infinitely many) numbers, called the terms of the sequence. There are a number of different ways to write a sequence. When n is unspecified, the expression an is called the general term of the sequence. Moreover, if we know that an = f(n) for some function f, then we say that f(n) is the formula for the general term. Common Sequences You have probably seen and worked with many different kinds of sequences already even if you didnt call them sequences. The natural numbers: 1, 2, 3, 4, 5, . The formula for the general term is very simple: an = n. The harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, . This is nothing more than the sequence of reciprocals of the natural numbers: an = 1/n. The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, . Here, the pattern is to start with two ones, and then to get each new term, we always add the previous two terms together. So 1 + 1 = 2, and then 1 + 2 = 3, and 2 + 3 = 5, and so on. The Fibonacci sequences is an example of a recursively-defined sequence, because we can write it by the following recursive rule. By the way, the Fibonacci sequence is important for many reasons, showing up in nature in the most unexpected ways. The nautilus shell grows in the shape of a logarithmic spiral, which is closely related to the Fibonacci sequence. Convergence and Divergence We say that a sequence converges to a number a if its terms get arbitrarily close to a the further along in the sequence you get. To be more precise, we say that the limit (as n ) of the convergent sequence exists (and equals a). Just like limits of functions, we use the lim notation. Convergence The Harmonic Sequence The harmonic sequence (an = 1/n) converges to 0. How can we establish this fact? Well intuitively speaking, if you plug in a very large value of n into the formula 1/n, what do you get? A little experimentation may lead you to the guess that 1/n converges to 0. If n = 10, you get 1/10 = 0.1, which is already pretty close to 0. For n = 100, you get 1/100 = 0.01, even closer to 0. For n = 1,000,000, the value is 1/(1,000,000) = 0.000001, very close to 0 indeed! The higher n is, the closer 1/n will get to 0. However to rigorously prove this requires a more careful argument. For our purposes, we will simply state this limit fact without proof. . Divergence The Natural Number Sequence If the limit does not exist, then we say that the sequence diverges (or is divergent). For example, the sequence of natural numbers, 1, 2, 3, 4, 5, . is divergent because the values simply get larger without bound. Theres no limit to the values, quite literally. Convergence and Limits As you can see from the definition, testing the convergence of a sequence requires taking a limit. There are a few standard tricks to working out these kinds of limits. The key is that the variable n is tending toward infinity (), so most of the same techniques that worked to find horizontal asymptotes also work in this new setting. By the way, now is a great time to review: How do you Find the Horizontal Asymptotes of a Function? Example Consider the sequence, . (a) Write out the first four terms of the sequence. (b) Determine whether the sequence converges. If it converges, then what is the limit? Solution (a) Each term of a sequence can be found by plugging in n-values, similar to how you might find the value of a function f(x) by plugging in x-values. Thus the first four terms are: 3, 2.75, 2, 1.3125. (b) We need to compute the limit as n approaches infinity. If you get stuck at this step, imagine how you would handle the problem if it involved x instead of n. For this one, Id use LHospitals Rule. The last line is based on the fact that the top of the fraction remains constant while the bottom grows without bound as n . Furthermore, we can now say that the sequence converges (because the limit does exists), and it converges to 0. Sequences are Not Series Now is a great time to remind you of a very important distinction in mathematical terminology. There is a related concept called a series, which is by definition the sum of a sequence. In fact, you may already know one important fact. Even though the harmonic sequence converges (to the value 0), the harmonic series actually diverges. In other words, the sum of all of the reciprocals of natural numbers just grows and grows without bound. For more about the distinction between sequences and series, as well as other topics on the AP Calculus BC exam, check out: AP Calculus BC Cram Sheet Summary While sequences make up only a small fraction of the material on an AP Calculus BC exam, they are very important to understand for other topics such as series. Here are the essential facts youll need to know. A sequence is a list of numbers, usually described by a pattern or formula. We call an the general term of the sequence, and if an = f(n), then f is a formula for the general term. A sequence (an) converges to the value a if the following limit statment is true: . Otherwise we say that the sequence diverges. A series is the sum of a sequence the two terms are not interchangeable.

Love and Desire Essaypilot

Drown, is a compilation of short stories, revolving around the themes of love and desire. In these stories, Drown Diaz addresses the tribulations and trials of Dominican immigrants as they seek opportunities to live the American Dream, after migrating to America. These stories are linked, although each piece functions as a separate independent work. Besides, they are narrated by a grown-up who is looking back at his childhood, in the setting of the 1980s. Another story is The Tattooer, which narrates a story of a youthful, talented tattooer named Seikichi, who was well-known for his sensual charm and unequaled boldness of his art as a secret desire, for instilling pain on men while they were under his needle. Seikichi likewise also desired to create an incredible masterpiece on the skin of a lovely woman. After years of patience, he eventually gets an opportunity to do so. Therefore, in this paper I will describe how love and desire is portrayed in these two stories. In the Drown, the story starts with Yunior as the narrator and his sibling Rafa. The two young men are sent to live with their uncle for the summer so that their mom can work long hours at a chocolate factory, in order to support them. Once more, it is uncovered that their father deserted the family when Yunior was only four years of age (Diaz 1). Afterwards, the family lived in destitution, to an extent of going without sustenance on occasions, so that the money could be spent on other basic needs such as clothing. Back then, Yunior was only eight and Rafa was twelve. When Yunior turned nine, his father Ramon abruptly returned to the Dominican Republic, and brought with him his entire family to the United States where he had been living. The family settled in New Jersey, where they still lived in destitution. he poverty here is not like that of the Dominican. In fact, Yunior brings out love by mentioning how the family though poor, it had food and other necessities, unlike in the Do minican Republic. The stories once again, focus on Ramons encounter in trying to live the American Dream after emigrating from the Dominican Republic. In fact, this story takes up an extensive part of the narrative, and the reader finds that when Ramon abandons his wife and kids, he arrives in the United States and starts working in Miami, after which he heads to New York. In New York, he finally gets married to a United States Citizen, in order to acquire permanent citizenship. He lives with the lady and the two even have a child together. Ramon through love for his family, left her, by going back to the Dominican Republic to get his family. For Yunior, his father returns after having left when he was only four years old. Yunior and his family are brought to the Unites States, where he now must start a new life. This change in place assists in highlighting how Yunior, as well as the rest of his family, desired chasing after the American Dream (Diaz 4). Once more, though the poverty in the United States was quite unlike the one he encountered in the Dominican Republic, Yuniors family was still poor, to an extent that he had to work while in high school to help his mother pay her bills. his condition reveals how sometimes dreams are far much bigger than reality. With time, he started selling marijuana and later he even got his own place after high school. He also had a special girlfriend. He too, thought of making something of himself, was his desire. He and Aurora imagined of having a house in the suburbs. This was however, shattered though, he acknowledges that he not only wanted to know how to make his dream come true, but he had also failed Aurora as a man, particularly when the story reveals that he had hit her earlier. In the end, Yunior is no different than the other men who mistreat her. Junichiro Tanizakis narrative, The Tattooer, begins with the storyteller describing the ancient art of tattooing. He perfectly exemplifies t hat Japanese men, who performed in the Kabuki Theater, received tattoos as a way of enhancing their beauty and satisfying their upper class audiences. This narrative however, is based on a young tattoo artist named Seikichi who was an ukiyoye painter in his youth but later dropped in social status and decided to become a well-known tattoo artist. For years, Seikichi advanced his tattoo creativity on several clients. According to him they were his body canvases which came in all various sizes and shapes, though he desired something more, perhaps a perfect canvas to paint his masterpiece on. one day, while passing close to a restaurant, he noticed a very beautiful ladys foot and fell in love with her. A couple of days later, the lovely woman showed up at his door step with a picture from one of Seikichis friends. He closely gazed at her and noticed the facial features that he had desired all along. There and then he knew that her body, was the perfect canvas he desired to paint his best masterpiece on. Unfortunately, the young woman was frightened by his gestures. Thus, he did not share his dreams. As much as he strained to persuade her, she still declined his offer to be his ultimate masterpiece (Tanizaki 3). Therefore, in order to get what he desired, he took hold of the the young woman and confined her. This section clearly puts forward the theme of desire.The following morning Seikichi began his perfect work of art on the sleeping lady. He did not stop till he completed his gem. Afterwards, the woman began to move about, the spider that he had tattooed on the ladys back moved as she did. His fine art was eventually alive and this gave him extraordinary delight. As the lady gradually gained her strength, she asked whether she could see the tattoo but he declined and made her bathe in warm water in order for the tattoo to bring out the colors. The warm however, water made her feel terrible pain since it made her skin sting. She shouted at him wait in the next room since, she did not want anyone to see her in so much pain. After an hour, the young woman emerged from the room dressed beautifully and with a twinkle in her eye. Seikichi was stunned at what he saw. She was lovely. Which expresses his love for her beauty. He gave her some artworks and requested her to leave but she declined. Seikichi made a request to see the tattoo once more and the woman slowly pivoted and removed kimono. A beam of light through the window caught the spider on her back, and it was engulfed in flames.Drugging the woman with a goal to satisfy his desire and by tattooing his best masterpiece on her back, the spider tattoo can be interpreted as a desired foretelling of a prophecy that was yet to come for the craftsman. Since the narrative does not specify what type of a spider was on her back, I would take a leap of faith and assume that the Seikichi drew a black widow spider. Nevertheless, a female black widow spider is a venomous, deadly creature that kills and consumes the male after mating. Meaning, if he did draw a female black spider on her back, it would clarify why the woman took on the physical appearance of a specific spider. The fact that they did not physically engage in any sexual act, as a reader I would incorporate the theme of love through tattooing aa a creative sexual act between the Seikichi and his canvas. It seems like she technically trapped him in her web of vengeance. First victim (Tanizaki 3). Along these lines, I would like to think that Seikichi let his love for art take control his reality and the effort of sacrificing everything to create his masterpiece. Though in the end, his own creation turns against him and he loses it all. If the spider was a black widow, then Seikichi sealed his fate if he did not know what the black widow spider meant. On the other hand, Diazs story has a loving theme to it, the rhythmic tempos give it a musical feel. Diaz achieves all this with casualness, ease and a very special voice. Therefore, the two major themes are clearly evident in the two stories. Works Cited Daz, Junot. Drown. 1996. New York: Riverhead (1997).Tanizaki, Junichiro. The Tattooer. Seven Japanese Tales. Trans. Howard Hibbett. New York: Knopf (1963): 136-43.