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Calculus:Series From Wikibooks Contents Introduction Convergence Absolute convergence Ratio test ... edit Introduction An arithmetic series is the sum of a sequence of terms. For example, an interesting series which appears in many practical problems in science, engineering, and mathematics is the geometric series r r r r where the indicates that the series continues indefinetly. A common way to study a particular series (following Cauchy) is to define a sequence consisting of the sum of the first n terms. For example, to study the geometric series we can consider the sequence which adds together the first n terms: Generally by studying the sequence of partial sums we can understand the behavior of the entire infinite series. Two of the most important questions about a series are Does it converge? If so, what does it converge to? For example, it is fairly easy to see that for r , the geometric series S n r will not converge to a finite number (i.e., it will diverge to infinity). To see this, note that each time we increase the number of terms in the series S n r increases.

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Calculus:Power series From Wikibooks The study of power series concerns ourselves with looking at series that can approximate some function over some interval. Contents Motivations Definitions Radius of convergence Differentiation and integration ... edit Motivations Recall from elementary calculus that we can obtain a line that touches a curve at one point by using differentiation. So in a sense we are getting an approximation to a curve at one point. This does not help us very much however. Let's look at the case of y =cos( x ), about the point x =0. We have a first approximation using differentiation by the line y =1. Observe that cos( x ) looks like a parabola upside-down at x =0 - so naturally we think what parabola could approximate cos( x ) at this point? 1- x edit Definitions A power series is a Series of the form a x a x a n x n edit Radius of convergence We can only use the equation to study f x ) when the power series converges. This may happen for a finite range, or for all real numbers. This interval in which the power series converges to the function is known as the radius of convergence E.g

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The UBC Calculus Online Homepage Welcome to UBC Calculus Online. This site is an online supplement to Math 101 being taught within the University of British Columbia Department of Mathematics. Everyone is welcome and feedback is appreciated. Who We Are Course Notes Labs Announcements In-Class Demonstrations Resources Links to some other interesting sites Please send us your comments.

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Langara College - Department of Mathematics and Statistics Internet Resources for the Calculus Student Topics in Calculus Precalculus Review Limits and Continuity Derivatives Approximation ... Other Other Internet Resources for Calculus and Analysis Tools Resource Collections, Courses and Programmes, If you have come across any good web-based calculus support materials that are not in the above listed collections, please do let us know and we may add them here. Give Feedback Return to Langara College Homepage

20. Sequences Tutorial On Sequences. Using Computer Programs To Plot Drill problems on geometric series, nth term test, and telescoping series. Tutorial on Comparison Test for testing convergence of series. http://archives.math.utk.edu/visual.calculus/6/

Sequences Tutorial on sequences. Using computer programs to plot the graph of a sequence. Also, using the TI-85 graphing calculator to plot sequences. Discussion [ Using Flash Using Java Using computer programs to generate a table of values of a sequence. Series Tutorial on introductory material on series. Interactive Javascript module on generating a table of values for geometric series. Interactive MathView notebook evaluating the partial sums of a geometric series. Drill problems on geometric series, n th term test, and telescoping series. Using computer programs to generate a table of values of partial sums. Using computer programs to plot the graph of the sequence of partial sums. Tutorial on the Integral Test. Error estimates using the Integral Test. Drill problems on using the integral test. Tutorial on Comparison Test for testing convergence of series. Tutorial on the Limit Comparison Test. Drill problems on using the limit comparison test. Tutorial on alternating series. Drill problems on using the alternating series test. Tutorial on the Ratio Test.

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