Science:Math Exam Resources/Courses/MATH105/April 2012/Question 02 (b)

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MATH105 April 2012

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Question 02 (b)
This problem contains three numerical series. For each of them, find out whether it converges or diverges. You should provide appropriate justification in order to receive credit. $\sum _{k=1}^{\infty }{\frac {k^{4}-2k^{3}+2}{k^{5}+k^{2}+k}}$

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
Compare the highest exponents of the numerator and the denominator to make an educated guess. Then confirm your guess using the limit comparison test.

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. 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

When we look at this series, we note that the leading power of the numerator is k4 and the leading power of the denominator is k5. Thus, for k very large, we would expect the terms in the series to look like k4/k5 = 1/k. We will use the limit comparison test with $a_{k}={\frac {k^{4}-2k^{3}+2}{k^{5}+k^{2}+k}}{\text{ and }}b_{k}={\frac {1}{k}}$ We compute ${\begin{aligned}\lim _{k\rightarrow \infty }{\frac {a_{k}}{b_{k}}}&=\lim _{k\rightarrow \infty }({\frac {k^{4}-2k^{3}+2}{k^{5}+k^{2}+k}})/({\frac {1}{k}})\\&=\lim _{k\rightarrow \infty }{\frac {k^{5}-2k^{4}+2k}{k^{5}+k^{2}+k}}\\&=\lim _{k\rightarrow \infty }{\frac {k^{5}/k^{5}-2k^{4}/k^{5}+2k/k^{5}}{k^{5}/k^{5}+k^{2}/k^{5}+k/k^{5}}}\\&=\lim _{k\rightarrow \infty }{\frac {1-2/k+2/k^{4}}{1+1/k^{3}+1/k^{4}}}\\&={\frac {1-0+0}{1+0+0}}=1\end{aligned}}$ As this limit is nonzero and finite, $\sum _{k=1}^{\infty }a_{k}$ converges if and only if $\sum _{k=1}^{\infty }b_{k}$ converges. The series $\sum _{k=1}^{\infty }b_{k}=\sum _{k=1}^{\infty }{\frac {1}{k}}$ diverges (it's worth remembering - or being able to show with the integral test - that $\sum _{k=1}^{\infty }{\frac {1}{k^{p}}}$ converges if $p>1$ and diverges if $p\leq 1$). The conclusion is that the series diverges!

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