21. ESAIM Control, Optimisation And Calculus Of Variations Part of European series in Applied and Industrial Mathematics. Full text from vol.1 (1995). http://www.edpsciences.com/cocv/

22. Visual Calculus - Series - 4 If the sequence sn is not convergent then we say that the series. is divergent. Properties of convergent series. Convergence of geometric series. http://archives.math.utk.edu/visual.calculus/6/series.4/

An example. Definition. Let n be a sequence. The n th partial sum s n , is the sum s n = a + a + ... + a n We obtain the sequence of partial sums n An interactive example illustrating the difference between the sequence of terms and the sequence of partial sums. Definition. Given a sequence n and the sequence of its partial sums s n , then we say that the series is convergent if the sequence s n is convergent and has finite limit. If then we write If the sequence s n is not convergent then we say that the series is divergent Properties of convergent series. Convergence of geometric series. Discussion [ Using Flash Using Java Example. Discussion [ Using Flash Using Java Divergence of harmonic series. Discussion [ Using Flash Using Java Theorem. n th Term Test. If the series is convergent, then Equivalently, if then the series is divergent. Discussion. Examples. [Discussion.] [Discussion.] Telescoping series. Examples. Discussion [ Using Flash Using Java Discussion [ Using Flash Using Java Drill Problems.

23. Lee Lady: Topics In Calculus A set of downloadable lectures. http://www.math.hawaii.edu/~lee/calculus/#Series-Sol

Topics in Calculus Professor Lee Lady University of Hawaii A Conceptual Approach to Applications of Integration The Derivatives of the Sine and Cosine Functions ... Review Problems for Differential Equations In my opinion, calculus is one of the major intellectual achievements of Western civilization - in fact of world civilization. Certainly it has had much more impact in shaping our world today than most of the works commonly included in a Western Civilization course books such as Descartes's Discourse on Method or The Prince by Machiavelli. But at most universities, we have taken this magnificent accomplishment of the human intellect and turned it into a boring course. Sawyer's little book What Is Calculus About? (Another book in the same vein, but more recent, is The Hitchhiker's Guide to Calculus by Michael Spivak.) For many of us mathematicians, calculus is far removed from what we see as interesting and important mathematics. It certainly has no obvious relevance to any of my own research, and if it weren't for the fact that I teach it, I would long ago have forgotten all the calculus I ever learned. But we should remember that calculus is not a mere ``service course.'' For students, calculus is the gateway to further mathematics. And aside from our obligation as faculty to make all our courses interesting, we should remember that if calculus doesn't seem like an interesting and worthwhile subject to students, then they are unlikely to see mathematics as an attractive subject to pursue further.

24. Calculus Notes calculus Notes. These notes (which are just an experiment for now) require a Look at the harmonic series example first if you are more interested in http://www.math.utah.edu/~carlson/teaching/calculus/

Calculus Notes Series The harmonic series The exponential series Home

25. Linear Methods This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. It is suitable for a first course on partial differential equations, Fourier series and special functions, and integral equations. Students are expected to have completed two years of calculus and an introduction to ordinary differential equations and vector spaces. Topics include Fourier series, Green's functions, and PDEs. http://www.mathphysics.com/pde/

Since your browser does not support frames, go here instead.

26. Infinite Series Since this limit exists, we say that the sum of the series is 2, even though we Now in fact the limit does exist, or, as we say, the series converges. http://www.math.utah.edu/~carlson/teaching/calculus/series.html

Infinite series An infinite series is an expression like this: S = 1 + 1/2 + 1/4 + 1/8 + ... The dots mean that infinitely many terms follow. We obviously can't add up an infinite number of terms, but we can add up the first n terms, like this: S S S S It is clear what the pattern is: the n-th partial sum is S n n When n gets larger and larger, S n gets closer and closer to the number 2. When a sequence S n gets closer and closer and closer to a given number S, we say that S is the limit of the S n 's and we write lim( S n ) = S To take a physical analogy, consider a student who is one yard from the wall of the classroom. He takes a large step to cut the distance to the wall in half. Then he takes another step to cut the distance in half again. He repeates this again and again, getting closer to the wall each time. He never reaches the wall, yet that is his limit postion. We could write lim( Position n ) = Wall In our case lim( S n ) = 2. Since this limit exists, we say that the sum of the series is 2, even though we can't really "do the sum." Another example Our first example was easy to understand because there is a simple formula for the partial sums. Now let's look at a more difficult example.

27. The Continental Mathematics League And The National Olympiads CML offers a series of math meets for grades 29, a calculus league for the high school, and a Java computer contest. http://www.continentalmathleague.hostrack.com/

Continental Mathematics League Welcome to the CML Math League Website. Below you will find information about our math league (click on "Math" below) and information about the National Science League, the National Social Studies League, the National Geography Challenge, the National Language Arts League, the Java Computer Contest, and the National Current Events League. If you have a home schooled student or need more information regarding any of the above Leagues, please contact us by phone at: or email us League Subject Continental Math League Math National Science League Science National Social Studies League Social Studies National Geography Challenge Geography National Language Arts League Language Arts National Current Events League Current Events Continental Mathematics League CML offers a series of math meets for grades 2-9, a calculus league for the high school, and a Java computer contest. Also available are seven books containing some of our best MATH questions. The books are: Best of CML Grades 4-9 covering 1980-1989 Best of CML Grades 2-3 covering 1990-1995 Best of CML Grades 4-6 covering 1990-1995 Best of CML Grades 7-9 covering 1990-1995 Best of CML Grades 2-3 covering 1996-2000 Best of CML Grades 4-6 covering 1996-2000 Best of CML Grades 7-9 covering 1996-2000 Best of CML Grades 2-3 covering 2000-2005 Best of CML Grades 4-6 covering 2000-2005 Best of CML Grades 7-9 covering 2000-2005 The cost of each book is $22 for league members and $30 for non-members. Below is information for the year 2005-2006 and some sample questions. Registration is now open for all CML offerings.Please call 631-584-2016 or fill out the CML Application Form to register or purchase books.

28. Convergence Tests For Infinite Series - HMC Calculus Tutorial so we urge you to look them up in your calculus text. For a particular series, one or more of the common convergence tests may be most convenient to http://www.math.hmc.edu/calculus/tutorials/convergence/

Convergence Tests for Infinite Series In this tutorial, we review some of the most common tests for the convergence of an infinite series k = a k = a + a + a The proofs or these tests are interesting, so we urge you to look them up in your calculus text. Let s a s a s n n k = a k n partial sums converges to a limit L, then the series is said to converge to the sum L and we write k = a k = L. For j k = a k converges if and only if k = j a k converges, so in discussing convergence we often just write a k Example Consider the geometric series k = x k The n th partial sum is s n = 1 + x + x + x n Multiplying both sides by x, xs n = x + x + x + x n+1 Subtracting the second equation from the first, (1-x)s n = 1-x n+1 so for x s n 1-x n+1 1-x For x lim n s n 1-x It is easy to see that k = x k diverges for x Thus k = x k = 1/(1-x) for x x Divergence Test If lim k a k 0, then k = a k diverges. Example The series k = k/(2k+1) diverges, since lim k k/(2k+1) = 1/2 Integral Test Let f(x) be continuous, decreasing, and positive for x Then k = 1 f(k) converges if and only if f(x)dx converges.

29. S.O.S. Math - Calculus Explains concepts in detail of limits, convergence of series, finding the derivative from the definition and continuity. Some basic formula conversions are given. http://www.sosmath.com/calculus/calculus.html

S.O.S. Homepage Algebra Trigonometry Differential Equations ... CyberBoard Search our site! S.O.S. Math on CD Sale! Only $19.95. Works for PCs, Macs and Linux. Tell a Friend about S.O.S. Books We Like Math Sites on the WWW S.O.S. Math Awards ... Privacy Concerns? We recommend: SEQUENCES Introduction to Sequences Basic Definitions Limit of a Sequence More on Limit of a Sequence ... Stirling's Formula SERIES Introduction to Series Convergence of Series The Geometric Series Application: A Bouncing Ball ... More Problems on Series LIMIT AND CONTINUITY Introduction and Basic Definitions Some Basic Properties The Pinching or Sandwich Theorem Limits and Infinity ... The Bisection Method DIFFERENTIATION The Definition of the Derivative Using the Definition to Compute the Derivative Techniques of Differentiation Derivatives of Trigonometric Functions ... More Problems on the Derivative INTEGRATION The Area Problem and the Definite Integral Properties of the Definite Integral More on the Area Problem The Fundamental Theorem of Calculus ... Mean Value Theorems for Integrals TECHNIQUES OF INTEGRATION Integration by Parts Integration of Rational Functions Substitution Trigonometric Substitution ... Other Trigonometric Powers Numerical Integration Problems on Techniques of Integration Integration of Nonelementary Functions LOCAL BEHAVIOR of FUNCTIONS Taylor Polynomials Indeterminate Forms: Introduction Indeterminate Quotient Forms Other Indeterminate Quotient Forms Improper Integrals Introduction and Basic Definitions Convergence and Divergence of Improper Integrals Tests of Convergence Absolute Convergence of Improper Integrals ... Problems on Improper Integrals Special Functions

30. Infinite Series/Calculus III - Physics Help And Math Help - Physics Forums Physics Forums is a science and physics community. http://physicsforums.com/showthread.php?t=84090

31. That's Calculus! Humorous approach to lessons , which are given using Real Player video clips. Topics cover the concept of limit, derivatives, and summation series. http://www.math.dartmouth.edu/~matc/eBookshelf/calculus/CalculusVideo/welcome.ht

"The introduction of limits causes much anxiety for many students in introductory calculus. This video is an excellent resource because it touches upon the important ideas underlying limits in a light-hearted, but correct way. It also serves a bridge enabling the instructor to pick up, review, and expand on this important concept. I highly recommend it." Richard Melka Professor of Mathematics University of Pittsburgh This humorous video review of basic calculus concepts is ideal to supplement high school or college mathematics classes, home schooling, or individual study. The series, featuring acclaimed performance artist Josh Kornbluth, was developed by Dartmouth College faculty for the Mathematics Across The Curriculum project with the support of the National Science Foundation. Episode 1: To The Limit. Josh explores the intuitive notion of limit from playground slides to food processors. Episodes 2 and 3: The Formal Limit. Josh plays the epsilon-delta game and visits the For All There Exists Cafe as he explains the formal definition of limit.

32. AcroTeX of a series of handson PDF files for students to explore calculus concepts. The demo is a calculus quiz (in PDF) users enter a multiple choice http://www.math.uakron.edu/~dpstory/acrotex.html

Educational Technical Material Demonstrations PDF Games Miscellaneous e-Calculus Publishing Mathematics on the Web: A Tutorial by K.K. Nambiar PDF Classroom Demonstrations Algeboard Algeboard/JS Akron Math Community Algebra Review in Ten Lessons ... Intuitive Set Theory: A Tutorial by K. K. Nambiar Using LaTeX to Create Quality pdf Documents Test/Quiz Grading Program FDF Frocessing Giants of Calculus ... WebTrig by Tom Price WhiteHole, BlackWhole, and The Book by K. K. Nambiar Enhancing Text in Text Fields using Pdfmarks JQuiz - a fill in the blank practice demo Flash Cards: Arithmetic ... Motion Mountain by Christoph Schiller Sample Exams: in Trig, Algebra and D.E. The AcroTeX eDucation Bundle FDF Processing Save Results to Web server Owl Adventures ... Probability Tutorials by Noel Vaillant Unix Environment and Programming in C by Marc van Dongen Web/Exerquiz, the French Versions PDF Calculator by Hans Hagen JJ_Game: Class Quizzes Inserting Doc Level JavaScript Updated Version for 5.0 at PlanetPDF ... Microeconomics by Dan Christiansen Mathematics Support Materials DIY: On Web Design, Hyperref, and Web/Exerquiz

33. Mr. Calculus Mr. calculus is available to give you hints on your calculus/mathematics questions. Contains links to AP resources, tables for derivatives, logarithms, exponentials, convergence tests for infinite series, and sequence limits. http://www.geocities.com/calculusisnumber1/

Ask Mr. Calculus Mr. Calculus is available to give you hints on your calculus/mathematics questions. Click "Ask Mr. Calculus".>Hints for mathematics subjects up to Calculus BC. Click above to go to Mr. Calculus page The 2004 AP Free Response solutions are linked on the Mr. Calculus page. Click above and then on the link on the Ask Mr. Calculus page.

34. Sequences And Series Equations - Calculus - EquationSheet Create a personal Equation Sheet from a large database of science and math equations including constants, symbols, and SI units. Large equation database http://www.equationsheet.com/sheets/Equations-47.html

We have a new Forum , please start posting away! Equations Constants SI Units Symbols ... LaTeX Utilities search for in Equation Description Equation LaTeX String Constant Description Constant Symbol Constant Value SI Unit Quantity SI Unit Name SI Unit Symbol SI Unit Equivalent Units Symbol Description ALL Use or for wildcard character, or to match any single character Welcome Guest MyEquationSheet Sign In Register Equations ... Sequences and Series Showing results to of , on page of DESCRIPTION EQUATION Arithmetic Series - Sequential Integers Arithmetic Series - Sequential Odd Integers Arithmetic Series - Sequential Squared Integers Cosine Definition as an Infinite Series Finite Geometric Series Infinite Geometric Series Sine Definition as an Infinite Series Terms of Service Bookmark Us

35. Calculus Equations - EquationSheet Equations calculus Showing results 1 to 10 of 52, on page 1 of 6 Arithmetic series Sequential Odd Integers, Arithmetic series - Sequential Odd http://www.equationsheet.com/sheets/Equations-2.html

We have a new Forum , please start posting away! Equations Constants SI Units Symbols ... LaTeX Utilities search for in Equation Description Equation LaTeX String Constant Description Constant Symbol Constant Value SI Unit Quantity SI Unit Name SI Unit Symbol SI Unit Equivalent Units Symbol Description ALL Use or for wildcard character, or to match any single character Welcome Guest MyEquationSheet Sign In Register Equations ... Calculus Showing results to of , on page of Differentiation/Differentals Integration/Integrals Limits Sequences and Series DESCRIPTION EQUATION Arithmetic Series - Sequential Integers Arithmetic Series - Sequential Odd Integers Arithmetic Series - Sequential Squared Integers Chain Rule of Differentiation Cosine Definition as an Infinite Series Definition of a Derivative Derivative of a Constant Derivative of a Natural Logarithm Derivative of a Variable to the First Power Derivative of a Variable to the nth Power Go to page: 1 Terms of Service Bookmark Us

36. Calculus.org - THE CALCULUS PAGE . SOS Math calculus Good list of problems with solutions. Through Fourier series. Ask Mr. calculus Ask Mr. calculus your questions. Some good problems. http://www.calculus.org/

37. Archive Of Reform Calculus Resources Some Good Cartoons (MAA Film series) for teaching Volumes (Stewart 6.2). A sample secondsemester calculus course with in-class groupwork activities, http://barzilai.org/archive/

Reform Calculus Resources Compiled by Harel Barzilai As featured on UTK Math Archives and CollegeBoard.com's APCentral (* and authored, where not otherwise noted) Activities Projects Capsules Resources [See also College Algebra (in preparation)] [Under construction: Black Mathematicians Listings indexed by section in Calculus: Concepts and Contexts by James Stewart. If there is enough interest, I will create other indexes. Planned: on-the-fly indexes by keyword. In-Class Activities (Stewart 0.0) Graph Your Story A Week in the Life of Sue Estimating the Mile Record Guidelines and Solutions (Preview of Calculus)(Functions) (Stewart 1.1) Estimating the Mile Record Guidelines and Solutions (Preview of Calculus)(Functions) (Stewart 1.2) Composing Functions (Composition of Functions) (Stewart 1.4) Air Traffic Controller (Polar)(Parametric) A Crash Course in Calculus (Polar)(Parametric) (Stewart 1.5) Generation E: Modeling Exponential Growth (Exponential Functions)(Exponential Identities)(Exponential Growth) (Stewart 2.1) The Car Trip Solutions to The Car Trip Designing a Speedometer and A Second Train (Tangents)(Velocities)(Rates of Change)(Derivative) (Stewart 2.5)

38. MAA Calculus Film Series MAA calculus Film series. We use The Volume of a Solid of Revolution (Chandler Davis, George Leger), which runs 8 minutes, and Volume by Shells (George http://barzilai.org/cr/maa-films.html

MAA Calculus Film Series We use The Volume of a Solid of Revolution Chandler Davis, George Leger ), which runs 8 minutes, and Volume by Shells George Leger ), which runs about 8 and a half minutes. These are the third-to-last and second-to-last films on Tape 3 , respectively, starting at 0:29:00 and 0:36:55 respectively on that tape. If there is extra time, the last film on Tape 3, Infinite Acres Melvin Henriksen ) is both thought-provoking for the students, and humorous. Note however that the film was produced several decades ago, and that there a couple of instances of "humor" between the husband and wife, and boss and employee for that matter, that would not be considered so funny today. Finally, you can put some the tapes on reserve in the library for the students to review before exams and during projects, even suggesting they can view some of the other videos to brush up on other material such as the Definite Integral, Fundamental Theorem of Calculus, etc.

39. Lee Lady: Topics In Calculus Taking differential calculus without studying differential equations is a lot like studying An Example of a series Solution of a Differential Equation http://www.math.hawaii.edu/~lee/calculus/

Topics in Calculus Professor Lee Lady University of Hawaii A Conceptual Approach to Applications of Integration The Derivatives of the Sine and Cosine Functions ... Review Problems for Differential Equations In my opinion, calculus is one of the major intellectual achievements of Western civilization - in fact of world civilization. Certainly it has had much more impact in shaping our world today than most of the works commonly included in a Western Civilization course books such as Descartes's Discourse on Method or The Prince by Machiavelli. But at most universities, we have taken this magnificent accomplishment of the human intellect and turned it into a boring course. Sawyer's little book What Is Calculus About? (Another book in the same vein, but more recent, is The Hitchhiker's Guide to Calculus by Michael Spivak.) For many of us mathematicians, calculus is far removed from what we see as interesting and important mathematics. It certainly has no obvious relevance to any of my own research, and if it weren't for the fact that I teach it, I would long ago have forgotten all the calculus I ever learned. But we should remember that calculus is not a mere ``service course.'' For students, calculus is the gateway to further mathematics. And aside from our obligation as faculty to make all our courses interesting, we should remember that if calculus doesn't seem like an interesting and worthwhile subject to students, then they are unlikely to see mathematics as an attractive subject to pursue further.

40. Calculus II (Math 2414) - Series & Sequences Online Notes / calculus II (Math 2414) / series Sequences and to be honest many students will never see series outside of their calculus class. http://tutorial.math.lamar.edu/AllBrowsers/2414/Seq_Series.asp

MPBodyInit('Seq_Series_files') Paul's Online Math Notes Sequences and Series In this chapter we’ll be taking a look at sequences and (infinite) series.  Actually this chapter will deal almost exclusively with series, however, we also need to understand some of the basics of sequences in order to properly deal with series.   We will therefore, spend a little time on sequences as well. Series is one of those topics that many students simply don’t see the use in studying and to be honest many students will never see series outside of their calculus class.  However, series do play an important role in the field of ordinary differential equations and without series large portions of the field of partial differential equations would not be possible.  In other words, series is an important topic even if you won’t ever see any of the applications.  Most of the applications are beyond the scope of most Calculus courses and tend to occur in classes that many students don’t take.  So, as you go through this material keep in mind that these do have applications even if we won’t really be covering many of them in this class. Here is a list of topics in this chapter.

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