(9/1/08)Math 10B. Lecture Examples.Section 9.4. Tests for convergence†Example 1 Does the series∞Xn=0(0.6)nn + 1converge?Answer:∞Xn=0(0.6)nn + 1converges by the Comparison Test with the convergent Geometric Series∞Xn=0(0.6)n.(All but the first partial sum of∞Xn=0(0.6)nn + 1in Figure A1a is less than the corresp onding partial sum of∞Xn=0(0.6)nin Figure A1b.)N10 20y12N10 20y12y =NXn=0(0.6)nn + 1y =NXn=0(0.6)nFigure A1a Figure A1bExample 2 Does∞Xn=110 cos(3n)n3/2converge?Answer:∞Xn=110 cos(3n)n3/2converges.Example 3 Does∞Xn=21n − 1converge or diverge?Answer:∞Xn=11n − 1diverges.Example 4 Does∞Xn=02n+ 105nconverge or diverge?Answer:∞Xn=02n+ 105nconverges by the Limit Comparison Test with the convergent geometric series∞Xn=025n.†Lecture notes to accompany Section 9.4 of Calculus by Hughes-Hallett et al1Math 10B. Lecture Examples. (9/1/08) Section 9.4, p. 2Example 5 Apply the Ratio Test to∞Xn=05nn!.Answer: The series converges.Example 6 Apply the Ratio Test to∞Xn=1(−2)nn3.Answer:∞Xn=1(−2)nn3diverges.Example 7 Show that the series∞Xn=1(−1)n+1√nconverges.Answer: The series converges by the Alternating Series Test.Example 8 Does∞Xn=1(−1)ne1/nconverge?Answer: No, the series diverges.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 10.4: Examples 1–5Section 10.5: Examples 1–5‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters an d section sof the textbook for the