HH_Sec_9_3

# HH_Sec_9_3 - Math 10B Lecture Examples Section 9.3...

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Unformatted text preview: (9/7/08) Math 10B. Lecture Examples. Section 9.3. Convergence of series† ∞ Example 1 Does n=2 ∞ 1 n(ln n)2 1 = converge or diverge? 1 • The improper integral and the inﬁnite series converge. (The ﬁrst 24 partial ln(2) 2 sums of the series are plotted in Figure A1.) Answer: x(ln x)2 y 2 y = sN N sN = n=2 1 n(ln n) 2 1 Figure A1 10 ∞ 20 N Example 2 Does n=1 ∞ 1 √ converge or diverge? n 1 Answer: 1 √ x dx = ∞ • The inﬁnite series diverges. (Its ﬁrst 25 partial sums are plotted in Figure A2.) y y = sN 8 N sN = n=1 1 √ n 4 Figure A2 10 ∞ 20 N Example 3 Does n=1 ∞ ne−n converge? ∞ 2 2 2 Answer: 1 xe−x dx = ∞ 1 −1 e 2 • ne−n converges. n=1 Example 4 Does n=1 ∞ 1 n 1 1.75 converge or diverge? Answer: n=1 † Lecture n 1.75 converges. notes to accompany Section 9.3 of Calculus by Hughes-Hallett et al 1 Math 10B. Lecture Examples. (9/7/08) ∞ Section 9.3, p. 2 Example 5 Does n=1 ∞ n converge or diverge? n+1 Answer: n=1 n diverges. n+1 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 10.3: Examples 1–4 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 ...
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HH_Sec_9_3 - Math 10B Lecture Examples Section 9.3...

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