Science:Math Exam Resources/Courses/MATH103/April 2016/Question 07 (a) (ii)

MATH103 April 2016

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Question 07 (a) (ii)
For each of the following series, decide whether or not it converges. Justify your answers. (ii) $\sum _{n=1}^{\infty }{\frac {n^{4}+3}{2n^{4}+n}}.$

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
Consider the divergence test.

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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. Note that $\lim _{n\to \infty }{\frac {n^{4}+3}{2n^{4}+n}}=\lim _{n\to \infty }{\frac {(n^{4}+3)/n^{4}}{(2n^{4}+n)/n^{4}}}=\lim _{n\to \infty }{\frac {1+3/{n^{4}}}{2+n/{n^{4}}}}={\frac {1+\lim _{n\to \infty }3/{n^{4}}}{2+\lim _{n\to \infty }1/{n^{3}}}}={\frac {1}{2}}\neq 0.$ Therefore, by the divergence test, the given series diverges.