Science:Math Exam Resources/Courses/MATH101/April 2018/Question 02 (iii)
• Q1 (i) • Q1 (ii) • Q1 (iii) • Q1 (iv) • Q2 (i) • Q2 (ii) • Q2 (iii) • Q3 (i) • Q3 (ii) • Q4 (i) • Q4 (ii) • Q5 (i) • Q5 (ii) • Q6 (i) • Q6 (ii) • Q7 (i) • Q7 (ii) • Q8 (i) • Q8 (ii) • Q9 (i) • Q9 (ii) • Q10 (i) • Q10 (ii) • Q10 (iii) • Q11 (i) • Q11 (ii) •
Question 02 (iii) 

For each of the following series, choose the appropriate statement. Write N, O, P, S, or T in each box. Each answer will be used at most once, and each series matches a single answer only.

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 

The easiest check for divergence of a series is if the summand does not tend to zero. 
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. The relevant integral for the first series is which can be computed using the substitution and shows that the series converges by the Integral Test. For the second series, observe that we have so the even terms of the limit tend to 4 and the odd terms to 4, which shows that the summand does not tend to zero (the limit does not even exist). This means that the series diverges.
For the third series, for any natural number , we have
so the series converges absolutely by the Comparison Test with a series.
For the fourth series, the Ratio Test gives
which shows that the series converges.
Answer: The respective answers are . 