![]() But if you have weaker ones, than it might make sense.and it's not crazy off the beaten path (see point 1). If you have strong CS students, would not bother, teach 'em new stuff. Note: That this is not a strong rationale, is a marginal one. And this is a part of that course that has some special relevance (more than partial fractions or the like). I'd say the rationale for giving them to CS AGAIN is that in general (IN GENERAL), CS are a bit math weak compared to other STEMS, so perhaps they need a bit more exposure to stuff they should have learned in standard second semester calculus (or even high school AP BC). ![]() but at the end of the second semester and usually a little harder, more "off the path" of just learning antidifferentiation tricks. Not even some "advanced calc" or "engine math" or God Help Us (real analysis) thingie. Are a normal part of the calculus sequence (even the AP BC) in the US. Rationale for (re)covering with CS students. ![]() Rationale for ever learning: They are related to numerical methods and algorithms. So many problems which may come up in modelling natural phenomena through computers can be treated quite easily be consider Taylor expansions. What does this mean? Suppose that I am a computer scientist and I wish to figure out how many terms of the series of $e$ I should take such that the difference between the series evaluation and actual value of $e$ below some error, then naturally the method of answering this systematically would be same as knowing what convergence is.Īnd more, the series convergence tests tells us what sequences it is worth to try search for bounds and which are not.Īnd to my understanding, Taylor series is like the swiss knife of modelling. 7.3: Mathematical Induction 7.3.1: Inductive Reasoning from Patterns 7.3.1.1: Inductive Proofs 7.3.2: Induction and Factors 7.3.3: Induction and Inequalities 7.4: Sums of Geometric Series 7.4.1: Sums of Finite Geometric Series 7.4.2: Sums of Infinite Geometric Series 7.5: Factorials and Combinations 7.5. Let make take for example convergent sequences, what does it mean to converge? If we have a sequence $(a_n)$ that means, for any possible $\epsilon$, there is some $N$ such that for any $m>N$, we have: Is there a textbook that has been used by previous instructors when teaching this class? If so, then you should be able to see how the text integrates these topics into the narrative, and where these topics get used to support other topics.īecause every single one of those concepts naturally come up when you want to "do" things with computers! One thing I don't quite understand about your list of topics is that you're supposed to be teaching them concepts taught in second-semester calculus (in the US), but the sub-topics include the integral test, and that makes it sound like they've already had a year of calculus. Whether one of your students will use Taylor series is probably luck of the draw, but they're a basic tool of literacy in STEM. Big O notation is a staple of computer science. Although it is also possible to define them without using limits, the style of those definitions is essentially the same as the style of definition used to define a limit: inequalities and several levels of quantifiers. In this case, it is possible to see Gibbs phenomenon at the end points of the interval.Big O and related notations relate closely to these notions. The figures below illustrate some partial Fourier series results for the components of a square wave.įunction s 6 ( x ). The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. ![]() The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. This application is possible because the derivatives of trigonometric functions fall into simple patterns. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. For example, it might be possible to organize gateway topics in biology, physics, chemistry, mathematics, and computer science into an integrated freshman. A Fourier series ( / ˈ f ʊr i eɪ, - i ər/ ) is an expansion of a periodic function into a sum of trigonometric functions.
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