Science:Math Exam Resources/Courses/MATH101/April 2015/Question 05 (a)
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Question 05 (a) |
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Determine whether the series is absolutely convergent, conditionally convergent, or divergent; justify your answer. |
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? |
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. |
Hint 1 |
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Recall the definition of absolute/conditional convergence. |
Hint 2 |
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To check whether the series absolutely converges, use either integral test or comparison test. |
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Solution |
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Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. 1. First we consider whether the series is absolutely convergent or not. Note that is a continuous, positive, and decreasing function on . On the other hand, the integral below diverges using the substitution :
This implies that by the integral test, the series diverges. i.e., it is not absolutely convergent. 2. Now we consider the conditional convergence. Since is positive, the given series is alternating series. Drawing its graph, it is easy to see that is monotonically decreasing and converges to 0. Therefore, by the alternating series test, we have .
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