Science:Math Exam Resources/Courses/MATH103/April 2010/Question 01 (d)

MATH103 April 2010

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Question 01 (d)
Multiple Choice Question: Exactly ONE of the answers provided is correct. There is no partial credit in this question. Which of the following series converges? i. $S=\sum _{k=1}^{\infty }k^{-2}.$ ii. $S=\sum _{k=1}^{\infty }{\frac {1}{k}}.$ iii. $S=\sum _{k=1}^{\infty }k.$ iv. $S=\sum _{k=0}^{\infty }(1.01)^{k}.$ v. $S=\sum _{k=0}^{\infty }k^{2}.$

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
A necessary (but not sufficient) condition that a series converges is that the summands go to zero. Which of the series above don't satisfy this criterion, and hence can not be convergent?

Hint 2
For the remaining candidates, use the p-series 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. 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. Let's first check if the summands of the offered series go to zero. Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \lim_{k \rightarrow \infty} k^{-2} &= \lim_{k \rightarrow \infty} \frac1{k^2} = 0. \\ \lim_{k \rightarrow \infty} \frac1{k} &= 0. \\ \lim_{k \rightarrow \infty} k &= \infty. \\ \lim_{k \rightarrow \infty} (1.01)^k &= \infty. \\ \lim_{k \rightarrow \infty} k^2 &= \infty. \end{align} } Thus iii., iv., and v. can not converge. But the series i. and ii. are candidates for convergent series. By a pairwise comparison of the summands of series i. and series ii. we see that k-2 < 1/k, and hence (Series i.) < (Series ii.). We are told that only one series converges, so it can only be Series i. (If Series ii. would converge, and (Series i.) < (Series ii.), then Series i. would also converge.) Indeed this is true, by the p-series test with p = 2. Thus, the correct answer is i. Fun fact: $\sum _{k=1}^{\infty }k^{-2}={\frac {\pi ^{2}}{6}}.$

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