Science:Math Exam Resources/Courses/MATH101/April 2016/Question 05 (b)

MATH101 April 2016

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Question 05 (b)
Does ${\displaystyle \sum _{n=1}^{\infty }{\frac {n^{4}2^{n/3}}{(2n+7)^{4}}}}$ converge or diverge? Justify your conclusion.

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
You can try the Limit Comparison Test, the Test for Divergence, or the Ratio Test.

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Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We apply the Limit Comparison Test with $a_{n}={\frac {n^{4}2^{n/3}}{(2n+7)^{4}}}\quad {\text{and}}\quad b_{n}=2^{n/3},$ which is valid since ${\begin{aligned}\lim _{n\rightarrow \infty }{\frac {a_{n}}{b_{n}}}=\lim _{n\rightarrow \infty }{\frac {n^{4}}{(2n+7)^{4}}}=\lim _{n\rightarrow \infty }{\frac {1}{{(2+7/n)}^{4}}}={\frac {1}{2^{4}}}\end{aligned}}$ exists and is nonzero. Since the series $\sum _{n=1}^{\infty }b_{n}$ is a divergent geometric series (with common ratio $r=2^{1/3}>1$), the given series diverges. The answer is diverge.

Solution 2
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. One could show $\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }{\frac {2^{n/3}}{(2+7/n)^{4}}}=\infty$ directly, and thus conclude that the given series diverges from the Test for Divergence.

Solution 3

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. one can apply the Ratio Test: ${\begin{aligned}\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|&=\lim _{n\to \infty }\left|{\frac {(n+1)^{4}2^{(n+1)/3}/(2(n+1)+7)^{4}}{n^{4}2^{n/3}/(2n+7)^{4}}}\right|\\&=\lim _{n\to \infty }{\frac {(n+1)^{4}(2n+7)^{4}}{n^{4}(2n+9)^{4}}}{\frac {2^{(n+1)/3}}{2^{n/3}}}=1\cdot 2^{1/3}>1.\end{aligned}}$ Therefore, the series diverges.