Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (b)
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Question 01 (b)  

Determine whether the following series converge or diverge. Check appropriate box.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Remember some critical rules we use in series. The divergence test, ratio test, series test, and limit comparison test are useful for this problem. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. (i) from ptest, p>1 then it is converge. (ii) We use the divergence test. First, compute . The limit can be written as
We have , and therefore
This implies that by the sequence is divergent by the divergence test. (iii) We use the ratio test. The ratio of the n+1 term to th n term is
we therefore need to compute
This simplifies to
The limit of
is 1, and the limit of 1/n is zero,
and the series converges by the ratio test. (iv) We will use the limit comparison test here. Letting be the sequence
and letting be the series
we have this is finite and nonzero, so the limit comparison test says that and both converge or both diverge. But diverges by the series test, so does too. '
