Question 07 (b)
Long Problem. Show your work. No credit will be given for the answer without the correct accompanying work.
Determine, with explanation, whether the following series converges or diverges.
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!
Evaluate for the first few values for to see a pattern.
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.
First, if we evaluate a few terms we see that , , , , and therefore the terms alternate. Thus, the given series is alternating series,
Since and is decreasing for , by the alternating series test, the series converges.