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Unformatted text preview: Math 231 Practice Exam 2 SOLUTIONS Problem 1: State as briefly and concisely as you can the following theorems/tests. List all hypothesis and the conclusions. Your statement should be formulated If (hypothesis) Then (conclusions). If it is possible for the test to fail so indicate. (i) Integral Test ( Hypothesis ) Suppose that f ( x ) is a positive, decreasing function and a n = f ( n ) . then ( Conclusion ) the series a n and the (improper)integral R 1 f ( x ) dx either both converge or both diverge. (ii) Limit Comparison Test Hypothesis Suppose that a n > for all n and lim a n b n = L > Then (Con- clusion) the series a n and b n either both converge or they both diverge. (iii) Comparison Test for SEQUENCES Hypothesis Suppose that < a n b n for all n (sufficiently large) Then (Conclusions) b n converges implies the series a n converges a n diverges implies the series b n diverges 1 (iv) k th Term Test There are two ways to state this: the way the test is stated in the book or the contrapositive, which is how it is usually used ( Hypothesis ) Suppose a n converges. then ( Conclusion ) lim n...
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