HH_Sec_10_3

HH_Sec_10_3 - (9/1/08) Math 10B. Lecture Examples. Section...

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Unformatted text preview: (9/1/08) Math 10B. Lecture Examples. Section 10.3. Finding and using Taylor series† ∞ Example 1 Use the Taylor series sin x = n=0 ∞ (−1)n 2n+1 x for y = sin x centered at x = 0 to (2n + 1)! give the Taylor series centered at x = 0 for y = sin(2x). Answer: sin(2x) = n=0 22n+1 (−1)n 2n+1 x (2n + 1)! Example 2 Use the Taylor series ∞ ln(1 + x) = n=1 (−1)n+1 n x = x − 1 x2 + 1 x3 − 1 x4 + · · · 2 3 4 n to find the Taylor series of f (x) = x2 ln(1 + x) centered at x = 0. ∞ Answer: x2 ln(1 + x) = n=1 (−1)n+1 n+2 x = x3 − 1 x4 + 1 x5 − 1 x6 + · · · 2 3 4 n ∞ Example 3 What is the Taylor series centered at x = 0 for y = f (x) if f (x) = n=0 1 xn for 2n + 10 −2 ≤ x < 2? ∞ Answer: f (x) = n=1 n 2 + 10 n xn−1 1 ∞ Example 4 Give an infinite series that equals 0 f (x) dx where f (x) = n=0 1 xn for 2 + 10 n −2 ≤ x < 2. 1 ∞ Answer: 0 f (x) dx = n=0 1 (n + 1)(2n + 10) Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 10.7: Examples 5–9 notes to accompany Section 10.3 of Calculus by Hughes-Hallett et al. chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. ‡ The † Lecture 1 ...
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