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Unformatted text preview: Sequences, Series John E. Gilbert, Heather Van Ligten, and Benni Goetz Why bother? So far in calculus weve spent a lot of time with functions: graphing them, taking derivatives, integrating, finding maxima and minima etc. For the polynomial 2 + 4 x- 6 x 2 + 7 x 3 , say, this was often particularly easy because we could deal with it term-term. For a function like e x , however, things were more complicated - we were even pretty vague about what the exponential number e = 2 . 71828 . . . meant! The crucial end-result was that d dx e x = e x , so e x cant be a polynomial, but perhaps we can approximate by a sequence of degree n polynomials a + a 1 x + a 2 x 2 + a 3 x 3 + . . . + a n x n , or maybe even represent e x as an infinite series e x = a + a 1 x + a 2 x 2 + a 3 x 3 + . . . + a n x n + . . . = n = 0 a n x n , for suitable choices of coefficients a , a 1 , a 2 , . . . by letting n . In other words, rewrite e x as a polynomial of infinite degree, so-to-speak, and make calculus for e x just as easy as for polynomials. This idea of representing complicated functions or approximating them using simple building blocks is extremely important in many applications of math to engineering, science, ... , music, CDs and digital signals quite generally! Lets start by approximating the graphs of e x and cos x shown in red with graphs of some as yet unspecified low-degree polynomials shown in blue and green: x y x y degree 1 degree 2 degree 2 degree 4 Notice how the chosen polynomial graphs approximate very smoothly near the origin . As the degree of the polynomial increases we might expect that the approximation gets better further away from the origin, and that when the degree of the polynomial is infinite the fit might be perfect everywhere!...
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