Science:Math Exam Resources/Courses/MATH220/December 2009/Question 07 (a)

MATH220 December 2009

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Question 07 (a)
Decide whether the following sequence converges or diverges. Prove your answers using the definition of convergence. $a_{n}=\cos \left({\frac {n\pi }{3}}\right).$

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
Start writing down the first few terms: $\displaystyle \cos(\pi /3)={\frac {1}{2}}$ $\displaystyle \cos(2\pi /3)={\frac {-1}{2}}$ $\displaystyle \cos(3\pi /3)=-1$ What will happen at $\displaystyle \cos(7\pi /3)$? Why? Can you use this to determine an answer?

Hint 2
Recall some important facts: If a sequence converges, it has one limit. If a sequence converges, then every subsequence must converge to that same limit. Can you find two subsequences that converge to different values thus showing that the sequence does not converge?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. This sequence diverges. To see this, we proceed by the hints. Consider the two subsequences defined by $\displaystyle a_{m}=\cos((1+6m)\pi /3)=\cos(\pi /3+2m\pi )$ $\displaystyle b_{m}=\cos((2+6m)\pi /3)=\cos(2\pi /3+2m\pi )$ By the $\displaystyle 2\pi$-periodicity of the cosine function, these two subsequences are constant sequences! In fact the first always equals a half ($a_{m}=1/2$ for all $m$) and the second is negative one half ($b_{m}=-1/2$ for all $m$). If our sequence converged, every subsequence would have to converge to the exact same value. Since we have found two subsequences that converge to different values, we have shown that our original sequence could not converge and hence diverges.

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