Binomial Theorem
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Sequences and Series Pg. 1
Sequences and Series Exercise Find the first four terms of the sequence 3 1, 1, 2, 3, . [Answer: 2, 5, 8, 11] Example Let be the sequence defined by 2, 1, and 21 13 2 for 0,1,2,, find and . Solution: (k = 0) 21 13 2 2 2 3 (k = 1) 32 24 2 6 2 8 ⁄ Exercise Without using a calculator, find the first 4 terms of the recursively defined sequence • 5, 2 3 [Answer: 5, 7, 11, 19] • 4, 1 [Answer: 4, , , ] • 2, 3, [Answer: 2, 3, 5, 8] Example Evaluate ∑ 0.5 and round your answer to 5 decimal places. ! Solution: ∑ 0.5 ! 0.5 0.5 0.5 ! ! ! 0.5 0.5 ! ! 0.5 0.5 0.5 0.5 0.5 0.5 0.041666 0.003125 0.000186 0.000009 . Exercise Find and evaluate each of the following sums • ∑ [Answer: 225] • ∑ 1 5 [Answer: 521] pg. 1 Sequences and Series Arithmetic Sequences an = an‐1 + d, also an = a1 + (n – 1)d n S = ()a +a n 2 1 n Exercise • For the arithmetic sequence 4, 7, 10, 13, (a) find the 2011th term; [Answer: 6,034] (b) which term is 295? [Answer: 98th term] (c) find the sum of the first 750 terms. [Answer: 845,625] rd th • The 3 term of an arithmetic sequence is 8 and the 16 term is 47. Find and d and construct the sequence. [Answer: 2,3;the sequence is 2,5,8,11,] • Find the sum of the first 800 natural numbers. [Answer: 320,400] • Find the sum ∑ 4 5. -
Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a . -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
3.3 Convergence Tests for Infinite Series
3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral. -
The Binomial Series 16.3
The Binomial Series 16.3 Introduction In this Section we examine an important example of an infinite series, the binomial series: p(p − 1) p(p − 1)(p − 2) 1 + px + x2 + x3 + ··· 2! 3! We show that this series is only convergent if |x| < 1 and that in this case the series sums to the value (1 + x)p. As a special case of the binomial series we consider the situation when p is a positive integer n. In this case the infinite series reduces to a finite series and we obtain, by replacing x with b , the binomial theorem: a n(n − 1) (b + a)n = bn + nbn−1a + bn−2a2 + ··· + an. 2! Finally, we use the binomial series to obtain various polynomial expressions for (1 + x)p when x is ‘small’. ' • understand the factorial notation $ Prerequisites • have knowledge of the ratio test for convergence of infinite series. Before starting this Section you should ... • understand the use of inequalities '& • recognise and use the binomial series $% Learning Outcomes • state and use the binomial theorem On completion you should be able to ... • use the binomial series to obtain numerical approximations & % 26 HELM (2008): Workbook 16: Sequences and Series ® 1. The binomial series A very important infinite series which occurs often in applications and in algebra has the form: p(p − 1) p(p − 1)(p − 2) 1 + px + x2 + x3 + ··· 2! 3! in which p is a given number and x is a variable. By using the ratio test it can be shown that this series converges, irrespective of the value of p, as long as |x| < 1. -
Sequences and Series
From patterns to generalizations: sequences and series Concepts ■ Patterns You do not have to look far and wide to fi nd 1 ■ Generalization visual patterns—they are everywhere! Microconcepts ■ Arithmetic and geometric sequences ■ Arithmetic and geometric series ■ Common diff erence ■ Sigma notation ■ Common ratio ■ Sum of sequences ■ Binomial theorem ■ Proof ■ Sum to infi nity Can these patterns be explained mathematically? Can patterns be useful in real-life situations? What information would you require in order to choose the best loan off er? What other Draftscenarios could this be applied to? If you take out a loan to buy a car how can you determine the actual amount it will cost? 2 The diagrams shown here are the first four iterations of a fractal called the Koch snowflake. What do you notice about: • how each pattern is created from the previous one? • the perimeter as you move from the first iteration through the fourth iteration? How is it changing? • the area enclosed as you move from the first iteration to the fourth iteration? How is it changing? What changes would you expect in the fifth iteration? How would you measure the perimeter at the fifth iteration if the original triangle had sides of 1 m in length? If this process continues forever, how can an infinite perimeter enclose a finite area? Developing inquiry skills Does mathematics always reflect reality? Are fractals such as the Koch snowflake invented or discovered? Think about the questions in this opening problem and answer any you can. As you work through the chapter, you will gain mathematical knowledge and skills that will help you to answer them all. -
Polar Coordinates and Complex Numbers Infinite Series Vectors And
Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms. -
Six Ways to Sum a Series Dan Kalman
Six Ways to Sum a Series Dan Kalman The College Mathematics Journal, November 1993, Volume 24, Number 5, pp. 402–421. Dan Kalman This fall I have joined the mathematics faculty at American University, Washington D. C. Prior to that I spent 8 years at the Aerospace Corporation in Los Angeles, where I worked on simulations of space systems and kept in touch with mathematics through the programs and publications of the MAA. At a national meeting I heard the presentation by Zagier referred to in the article. Convinced that this ingenious proof should be more widely known, I presented it at a meeting of the Southern California MAA section. Some enthusiastic members of the audience then shared their favorite proofs and references with me. These led to more articles and proofs, and brought me into contact with a realm of mathematics I never guessed existed. This paper is the result. he concept of an infinite sum is mysterious and intriguing. How can you add up an infinite number of terms? Yet, in some contexts, we are led to the Tcontemplation of an infinite sum quite naturally. For example, consider the calculation of a decimal expansion for 1y3. The long division algorithm generates an endlessly repeating sequence of steps, each of which adds one more 3 to the decimal expansion. We imagine the answer therefore to be an endless string of 3’s, which we write 0.333. .. In essence we are defining the decimal expansion of 1y3 as an infinite sum 1y3 5 0.3 1 0.03 1 0.003 1 0.0003 1 . -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
3.2 Introduction to Infinite Series
3.2 Introduction to Infinite Series Many of our infinite sequences, for the remainder of the course, will be defined by sums. For example, the sequence m X 1 S := : (1) m 2n n=1 is defined by a sum. Its terms (partial sums) are 1 ; 2 1 1 3 + = ; 2 4 4 1 1 1 7 + + = ; 2 4 8 8 1 1 1 1 15 + + + = ; 2 4 8 16 16 ::: These infinite sequences defined by sums are called infinite series. Review of sigma notation The Greek letter Σ used in this notation indicates that we are adding (\summing") elements of a certain pattern. (We used this notation back in Calculus 1, when we first looked at integrals.) Here our sums may be “infinite”; when this occurs, we are really looking at a limit. Resources An introduction to sequences a standard part of single variable calculus. It is covered in every calculus textbook. For example, one might look at * section 11.3 (Integral test), 11.4, (Comparison tests) , 11.5 (Ratio & Root tests), 11.6 (Alternating, abs. conv & cond. conv) in Calculus, Early Transcendentals (11th ed., 2006) by Thomas, Weir, Hass, Giordano (Pearson) * section 11.3 (Integral test), 11.4, (comparison tests), 11.5 (alternating series), 11.6, (Absolute conv, ratio and root), 11.7 (summary) in Calculus, Early Transcendentals (6th ed., 2008) by Stewart (Cengage) * sections 8.3 (Integral), 8.4 (Comparison), 8.5 (alternating), 8.6, Absolute conv, ratio and root, in Calculus, Early Transcendentals (1st ed., 2011) by Tan (Cengage) Integral tests, comparison tests, ratio & root tests. -
A Tentative Interpretation of the Epistemological Significance of the Encrypted Message Sent by Newton to Leibniz in October 1676
Advances in Historical Studies 2014. Vol.3, No.1, 22-32 Published Online February 2014 in SciRes (http://www.scirp.org/journal/ahs) http://dx.doi.org/10.4236/ahs.2014.31004 A Tentative Interpretation of the Epistemological Significance of the Encrypted Message Sent by Newton to Leibniz in October 1676 Jean G. Dhombres Centre Alexandre Koyré-Ecole des Hautes Etudes en Sciences Sociales, Paris, France Email: [email protected] Received October 8th, 2013; revised November 9th, 2013; accepted November 16th, 2013 Copyright © 2014 Jean G. Dhombres. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Jean G. Dhombres. All Copyright © 2014 are guarded by law and by SCIRP as a guardian. In Principia mathematica philosophiæ naturalis (1687), with regard to binomial formula and so general Calculus, Newton claimed that Leibniz proposed a similar procedure. As usual within Newtonian style, no explanation was provided on that, and nowadays it could only be decoded by Newton himself. In fact to the point, not even he gave up any mention of Leibniz in 1726 on the occasion of the third edition of the Principia. In this research, I analyse this controversy comparing the encrypted passages, and I will show that at least two ideas were proposed: vice versa and the algebraic handing of series. Keywords: Leibniz; Newton; Calculus Principia; Binomial Series; Controversy; Vice Versa Introduction tober 1676 to Leibniz. -
Calculus Online Textbook Chapter 10
Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms.