HH_Sec_9_4 - (9/1/08) Math 10B. Lecture Examples. Section...

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Unformatted text preview: (9/1/08) Math 10B. Lecture Examples. Section 9.4. Tests for convergence† ∞ Example 1 Does the series n=0 ∞ (0.6)n converge? n+1 ∞ Answer: n=0 (0.6) n n+1 converges by the Comparison Test with the convergent Geometric Series ∞ n=0 (0.6)n . ∞ (All but the first partial sum of n=0 (0.6) n n+1 in Figure A1a is less than the corresponding partial sum of n=0 (0.6)n in Figure A1b.) y y 2 1 2 1 10 N 20 (0.6)n n+1 N 10 N 20 (0.6)n n=0 N y= n=0 y= Figure A1a ∞ Figure A1b Example 2 Does n=1 ∞ 10 cos(3n) n3/2 n3/2 converge? Answer: n=1 10 cos(3n) converges. ∞ Example 3 Does n=2 ∞ 1 converge or diverge? n−1 1 diverges. Answer: n=1 n−1 ∞ Example 4 Does n=0 ∞ 2n + 10 converge or diverge? 5n ∞ Answer: n=0 2n + 10 5n converges by the Limit Comparison Test with the convergent geometric series n=0 2n . 5 † Lecture notes to accompany Section 9.4 of Calculus by Hughes-Hallett et al 1 Math 10B. Lecture Examples. (9/1/08) ∞ Section 9.4, p. 2 5n . n! (−2)n n3 Example 5 Apply the Ratio Test to n=0 Answer: The series converges. ∞ Example 6 Apply the Ratio Test to n=1 ∞ . Answer: n=1 (−2)n n3 diverges. ∞ Example 7 Show that the series Example 8 n=1 Answer: The series converges by the Alternating Series Test. ∞ n 1/n (−1)n+1 √ converges. n Does n=1 Answer: No, the series diverges. (−1) e converge? Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 10.4: Examples 1–5 Section 10.5: Examples 1–5 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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This note was uploaded on 09/13/2010 for the course MATH math 10b taught by Professor Reed during the Summer '10 term at UCSD.

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HH_Sec_9_4 - (9/1/08) Math 10B. Lecture Examples. Section...

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