Science:Math Exam Resources/Courses/MATH103/April 2013/Question 01 (b)

MATH103 April 2013

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Question 01 (b)
Sequences and Series: Short Answer Problems - Determine whether the following sequences and series converge (full marks for correct answer with justification; work must be shown for partial marks) Does the sequence $\left({\frac {n^{3}+2}{{\sqrt {n^{6}+3}}+{\sqrt[{3}]{27n^{9}+1}}}}\right)_{n\geq 1}$ converge? If it does, calculate the limit.

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
Don't let the square and cube roots fool you! This is still essentially a polynomial divided by a polynomial so factor out the largest power of n in both the numerator and the denominator.

Hint 2
What are the dominant terms when $n$ is large?

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

Proceeding as in the hint, we have ${\begin{aligned}{\frac {n^{3}+2}{{\sqrt {n^{6}+3}}+{\sqrt[{3}]{27n^{9}+1}}}}&={\frac {n^{3}(1+{\tfrac {2}{n^{3}}})}{{\sqrt {n^{6}(1+{\tfrac {3}{n^{6}}})}}+{\sqrt[{3}]{27n^{9}(1+{\tfrac {1}{27n^{9}}})}}}}\\&={\frac {n^{3}(1+{\tfrac {2}{n^{3}}})}{n^{3}{\sqrt {(1+{\tfrac {3}{n^{6}}})}}+3n^{3}{\sqrt[{3}]{(1+{\tfrac {1}{27n^{9}}})}}}}\\&={\frac {1+{\tfrac {2}{n^{3}}}}{{\sqrt {(1+{\tfrac {3}{n^{6}}})}}+3{\sqrt[{3}]{(1+{\tfrac {1}{27n^{9}}})}}}}\end{aligned}}$ Taking the limit as n tends to infinity in the last equality above yields $\lim _{n\to \infty }{\frac {n^{3}+2}{{\sqrt {n^{6}+3}}+{\sqrt[{3}]{27n^{9}+1}}}}=\lim _{n\to \infty }{\frac {1+{\tfrac {2}{n^{3}}}}{{\sqrt {(1+{\tfrac {3}{n^{6}}})}}+3{\sqrt[{3}]{(1+{\tfrac {1}{27n^{9}}})}}}}={\frac {1}{1+3}}={\frac {1}{4}}$ and so the sequence converges to one fourth.

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