Likewise, searching up relevant terms in the subreddit search bar can also answer your question for you. Make sure you first look up definitions you don't know, instead of asking us. "High school problem" and "help plz" are not good titles. (Note: This rule is relaxed for extracurricular problems and questions of clarification.)ĭescriptive titles. In other words, show your numbers, if applicable. Just saying you made an attempt, or how you made an attempt, is not considered proof of working. Showing only your final answer is not considered proof of working. Posts that do not obey this rule will be removed until proof of working is shown. Try SOMETHING before you post here, and post your attempts. RULES FOR SUBMISSIONS:ĭO NOT ASK US TO HELP YOU ON TESTS OR EXAMS. If you're willing to learn, we're willing to teach.įull rules and FAQ are here. r/MathHelp is, as the tin says, help with your mathematical questions. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License Continuing this way, find a formula for the expected number of top random insertions needed to consider the deck to be randomly shuffled. The two cards below B B are now in random order. The expected number of top random insertions before this happens is n / 2. Once one card is below B, B, there are two places below B B and the probability that a randomly inserted card will fall below B B is 2 / n. Thus the expected number of top random insertions before B B is no longer at the bottom is n. If the deck has n n cards, then the probability that the insertion will be below the card initially at the bottom (call this card B ) B ) is 1 / n. We will consider a deck to be randomly shuffled once enough top random insertions have been made that the card originally at the bottom has reached the top and then been randomly inserted. The simplest way to shuffle cards is to take the top card and insert it at a random place in the deck, called top random insertion, and then repeat. In Figure 5.12, we depict the harmonic series by sketching a sequence of rectangles with areas 1, 1 / 2, 1 / 3, 1 / 4 ,… 1, 1 / 2, 1 / 3, 1 / 4 ,… along with the function f ( x ) = 1 / x. To illustrate how the integral test works, use the harmonic series as an example. It is important to note that this test can only be applied when we are considering a series whose terms are all positive. This test, called the integral test, compares an infinite sum to an improper integral. This technique is important because it is used to prove the divergence or convergence of many other series. In this section we use a different technique to prove the divergence of the harmonic series. In the previous section, we determined the convergence or divergence of several series by explicitly calculating the limit of the sequence of partial sums and showing that S 2 k > 1 + k / 2 S 2 k > 1 + k / 2 for all positive integers k. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 5.3.2 Use the integral test to determine the convergence of a series.5.3.1 Use the divergence test to determine whether a series converges or diverges.
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