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- mastermathmentor.com presents Helping students learn and teachers teach Created by: Stu Schwartz Un-narrated Version Graphics: Apple Grapher: Version 2.3 Math Type: Version 6.7 Intaglio: 2.9.5a Fathom: Version 2.11
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- Calculating Pi www.mastermathmentor.com We know that the circumference C of a circle is calculated by multiplying times the diameter. So is the ratio of the circumference to the diameter. But how do we calculate the value of without this geometric approach? Circumference diameter
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- www.mastermathmentor.com Terms used in this slide show: Sequence p-series test Recursive formula Harmonic series Explicit formula Alternating series test Convergent sequence Telescoping series Divergent sequence Integral test Series Direct comparison test Infinite series Limit comparison test Convergent series Ratio test Divergent series Absolutely convergent Convergence test Conditionally convergent n th term test Root test Geometric test Vocabulary
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- Sequences www.mastermathmentor.com We define a sequence as a set of numbers that has an identified first member, 2 nd member, 3 rd member, and so on. We use subscript notation to denote sequences: a 1, a 2, a 3, etc. Sequences are rarely made up of random numbers; they usually have a pattern to them. What is the 4 th and 5 th members of the sequence: 1, 2, 4, ? If you said that the 4 th member is 8 and the 5 th member is 16, the pattern is: to get the next term, we double the previous term. If you said that the 4 th member is 7 and the 5 th member is 11, the pattern is: we add 1 to the first member, add 2 to the 2 nd member and thus add 3 to the 3 rd member. The problem with giving the terms of a sequence is that there is an assumption that the student can see the pattern of the terms. Sometimes the pattern isnt obvious, even if many terms are given. For instance, finding the next term in the sequence 1, 2, 3/2, 2/3, 5/24 is quite difficult, and yet when you see the answer in the next slide, it makes perfect sense.
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- Defining Sequences www.mastermathmentor.com Recursive: a 1 = 1, a n = 2a n-1 The 5 th term would be 16. Advantage: pattern easy to see. Disadvantage: to find the 10 th term, we need the 9 th which needs the 8 th, For the sequence: 1, 2, 4, 8, Rather than give the terms of a sequence, we give a formula for the nth term a n. There are two ways to do so: recursively and explictly. Recursive: we give the first term a 1 and the formula for the n th term a n in terms of a function of the (n - 1) st term, a n-1. Explicit: we give a formula for the n th term a n in terms of n. Explicit: a n = (1/6)n 3 - (1/2)n 2 + (4/3) The 5 th term would be 15. Advantage: easy to find the 10 th term. Disadvantage: formula is not obvious. In this course, students rarely will be asked to generate a formula for the n th term from the terms themselves. However, if they are given the n th term formula, which is usually given explicitly, they should be able to find any term. For instance, the sequence 1, 2, 3/2, 2/3, 5/24, is defined as a n = n 2 /n!. The 6 th term is 36/720 = 1/20.
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- Convergent and Divergent Sequences www.mastermathmentor.com Sequences either converge (are convergent) or diverge (are divergent). Convergent sequences: the limit of the n th term as n approaches exists Divergent sequences: the limit of the n th term as n approaches does not exist
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- Using LHospitals Rule www.mastermathmentor.com Whenever you are asked about the convergence or divergence of a sequence when given a n, it is best to write out a few terms of the sequence to get a sense of it. Even then, you can get fooled. To be sure, you can use LHospitals rule to find the limit of a n as n approaches infinity, and if LHospitals rule cannot be used, use simple logic.
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- Series www.mastermathmentor.com We define a series as the sum of the members of a sequence starting at the first term and ending at the n th term: We define an infinite series as the sum of the members of a sequence starting at the first term and never ending. The remainder of this slide show is centered whether an infinite series converges (has a limit). A sequence converges if its n th term a n has a limit as n approaches infinity. A series converges if the sum of its terms has a limit as n approaches infinity. Determine whether each of the following series converges.
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- Convergence Tests www.mastermathmentor.com A convergence test is a procedure to determine whether an infinite series given the formula for a n is convergent. Some convergence tests can be done by inspection while others involve a bit of work that will need to be shown. The n th term test If the n th term does not converge to zero, the series must diverge. If the n th term does converge to zero, the series can converge. the sequence terms the partial series
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- Geometric Series & Convergence www.mastermathmentor.com A series in the form of is a geometric series. The Geometric test If |r| 1, the series diverges. If |r| < 1, the series converges to a/(1 r) The figure is a square of side 8 and the midpoints of the square are vertices of an inscribed square with the pattern continuing forever. Show that the sum of the areas and perimeters are convergent.
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- The p-Series and Harmonic Series www.mastermathmentor.com A series in the form of where p > 0, is a p-series. The p-series test: If 0 < p 1, the series diverges. If p > 1, the series converges. p is a positive constant: Any series in the form of [(c/an + b)] is called an harmonic series and is divergent. the sequence terms the partial series
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- More on the Harmonic Series 1/n www.mastermathmentor.com The length of the overhang is where n is the number of books. This is the harmonic series and thus theoretically, it will balance with an infinite number of books. Suppose we stack identical books of length 1 so that the top book overhangs the book below it by , which overhangs the book below it by , which overhangs the book below it by , etc. This structure, called the Leaning Tower of Lire will (just barely) balance. 1/10 It takes 31 books for the overhang to be 2 books long, 227 books for the overhang to be to be 3 books long, 1,674 books for the tower to be 4 books long, and over 272 million books for the overhang to be 10 books long. Try it with a deck of cards. Deck of 52 cards It seems counter-intuitive that is divergent. Heres a simple proof:
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- Alternating Series www.mastermathmentor.com An alternating series is one whose terms alternate in signs. The Alternating series test: Error in an alternating series In a convergent alternating series, the error in approximating the value of the series using N terms is the (N + 1) st term.
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- Telescoping Series www.mastermathmentor.com If we take a divergent series and subtract a divergent series, we can get another divergent series. However, it is possible that we subtract one divergent series from another, our answer converges. A telescoping series is an alternating series in the form of. If this passes the n th term test, the series converges. Expand the expression.
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- The Integral Test www.mastermathmentor.com If the integral test shows convergence of a series, the value of the integral is not the value of the series. It is merely an indicator that the series converges.
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- Comparison Tests www.mastermathmentor.com The Direct Comparison test: If there is a deviation from the forms already studied, the tests cannot be used.
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- The Limit Comparison Test www.mastermathmentor.com The Limit Comparison test:
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- The Ratio Test www.mastermathmentor.com The Ratio test:
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- Failure of the Ratio Test www.mastermathmentor.com The ratio test will always be inconclusive with a series in the form of a polynomial over a polynomial or polynomials under radicals. a n absolutely convergent: a n converges and |a n | converges a n conditionally convergent: a n converges and |a n | diverges You may wonder: if the ratio test is so versatile, why do we need all of the other tests? Why not just apply the ratio test immediately?
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- The Root Test www.mastermathmentor.com The following numbers are written from smallest to largest: n 20, 20 n, n!, n n. While an expression of a smaller expression over a larger expression could converge, an expression of a larger expression over a smaller expression diverges. The Root test:
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- Series Convergence/Divergence Flowchart www.mastermathmentor.com n th term test No Yes Geometric test Yes No p-series test Yes No Alternating series test Yes No Is the series telescoping? Yes No
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- More Tests in Order of Usefulness www.mastermathmentor.com Limit Comparison test Yes No Ratio test Yes No Root test Yes No Integral test Yes No
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- Vocabulary www.mastermathmentor.com Do you understand each term? Sequence p-series test Recursive formula Harmonic series Explicit formula Alternating series test Convergent sequence Telescoping series Divergent sequence Integral test Series Direct comparison test Infinite series Limit comparison test Convergent series Ratio test Divergent series Absolutely convergent Convergence test Conditionally convergent n th term test Root test Geometric test