Math191_Week14_section - Math 191 Problem Set for Week 13 Section Theorem 0.1(The nth Term Test If an fails to exist or is different from zero then n=1

# Math191_Week14_section - Math 191 Problem Set for Week 13...

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Math 191 Problem Set for Week 13 Section Theorem 0.1 (The nth Term Test) If a n fails to exist or is different from zero, then X n =1 a n diverges. Theorem 0.2 (The Integral Test) Let { a n } be a sequence of positive terms. Suppose that a n = f ( n ) , where f ( x ) is a continuous, positive, decreasing function of x for all x N ( N a positive integer). Then the series X n = N a n and the integral R N f ( x ) dx both converge or both diverge. Theorem 0.3 (The Direct Comparison Test) Let a n be a series with no negative terms. a. a n converges if there is a convergent series c n with a n c n for all n > N , for some integer N . b. a n diverges if there is a divergent series d n with a n d n for all n > N , for some integer N . Theorem 0.4 (The Limit Comparison Test) Suppose that a n > 0 and b n > 0 for all n N ( N an integer). a. If lim n →∞ a n b n = c > 0 , then a n and b n both converge or both diverge. b. If lim n →∞ a n b n = 0 and b n converges, then a n converges.