1 10.7: Taylor SeriesGoal: Given any differentiable function f, find a power series∞Xn=0cn(x − a)nwhich is equal to f (x)for all x on its interval of convergence.Easier Look: Let a = 0 (This is also called a Maclaurin Series). We begin by assuming the firstequation below is true for all x on the interval o f convergence of the series:f(x) =∞Xn=0cnxn= c0+ c1x + c2x2+ c3x3+ · · ·1If TNis the Nth partial sum of the Taylor series of f (a lso called the nth-degree Taylor Polynomialof f), and RNis the remainder (i.e., the sum of the rest of the terms), we can prove the series convergesto f (x) (for all x in the interval of convergence) by proving RN(x) → 0.Example: Find the Maclaurin Series for f (x) = sin x and explain why the serie s converges to sin x.2As we can see, it is easy to estimate RN(x) if the Taylor series is alternating (using the AlternatingSeries Estimation Theore m). If not, we can use the following:Taylor’s Inequality: If |fN +1(x)| ≤ M for |x − c| ≤ r, then the remainder RN(x) of the N th degreeTaylor polynomial of f satisfies the inequalityExample: Find the Taylor ser ies for f(x) = excentered at x = 2. You do not have to prove tha tRN→ 0.3IMPORTANT MACLAURIN SERIES TO KNOW:1. ex=2.11 − x=3. sin x =Use a Taylor Series to writeˆx0e−t2dt as a power series.4Some interesting results of Taylor
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