Science:Math Exam Resources/Courses/MATH101/April 2016/Question 02 (a)
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Question 02 (a) 

Only one of the following statements is always true; determine which one is true. (Assume and are continuous functions.) A: If , then converges. B: If is an odd function, then . C: If , then converges. D: If for all and converges, then converges. E: 
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Hint 

For A and C, consider the fact that diverges. For B, try to compute the integrals for some odd function. (for example, ) For D, recall the comparison test. For E, try the substitution . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. A: False. A counter example is . B: False. Try . Then In fact, using substitution , . Note that because is an odd function. C: False. A counter example is . D: True. Since converges, this integral has certain value(number), say . Then the assumption on implies that
by the comparison property of integral. Therefore, , so that also has a certain value between 0 and M. i.e., convergent. E: False. A counter example is . , while . In fact, using substitution , . The answer is D. 