# n2 - x on a longer and longer interval about 0 To be...

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MATH 153 Note 2: Graphical Presentation of a Taylor Series To convince you that the Taylor series for sin x centered at x = 0 does in fact converge to sin x , and to more convincingly explain to you how that convergence happens, I’m going to draw pretty pictures. Recall the following: sin x = X n =0 ( - 1) n (2 n + 1)! x 2 n +1 Let’s start with a picture of what we’re after: Figure 1. The function sin x . The ﬁrst interesting Taylor polynomial we have uses just the ﬁrst term: T 1 ( x ) = x. Note that this is just the tangent line to sin x at the center of the series, x = 0—a function we’ve claimed since 151 approximates sin x well near x = 0. Figure 2. sin x and T 1 ( x ) 1

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2 We don’t really have a T 2 ( x ); the coeﬃcient of x 2 in the sin x power series is 0, so T 2 ( x ) = T 1 ( x ). We do have T 3 ( x ): T 3 ( x ) = x - x 3 3! . Figure 3. sin x and T 3 ( x ) A couple more: T 5 ( x ) = x - x 3 3! + x 5 5! . Figure 4. sin x and T 5 ( x ) T 7 ( x ) = x - x 3 3! + x 5 5! - x 7 7! .
3 Figure 5. sin x and T 7 ( x ) Notice how these polynomials more and more closely approximate sin

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Unformatted text preview: x on a longer and longer interval about 0. To be extreme about it, let’s skip ahead to T 25 ( x ): T 25 ( x ) = x-x 3 3! + x 5 5!-x 7 7! + x 9 9!-x 11 11! + x 13 13!-x 15 15! + x 17 17!-x 19 19! + x 21 21!-x 23 23! + x 25 25! . Figure 6. sin x and T 25 ( x ) To the naked eye, T 25 ( x ) lies right on top of sin x for three periods of sin x ! Another way to think of this would be to consider the remainder functions R n ( x ) = sin( x )-T n ( x ). Let’s graph them all on one set of axes; notice how each one seems to lie on the x-axis for longer than the previous ones (i.e., that R n ( x ) → 0 for all x ): 4 Figure 7. Plots (from inside out) of R 1 ( x ) (solid), R 3 ( x ) (dot-ted), R 5 ( x ) (dashed), R 7 ( x ) (dot-dashed) and R 25 ( x ) (solid again)...
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n2 - x on a longer and longer interval about 0 To be...

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