Comparison Tests Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{5\ln{n}}{n^7}$ converge?

   (a) Converges by comparison with $\sum \dfrac{5}{n^6}$
   (b) Converges by comparison with $\sum \dfrac{5}{n^8}$
   (c) Diverges by comparison with $\sum \dfrac{5}{n^6}$
   (d) Diverges by comparison with $\sum \dfrac{5}{n^8}$

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2. Does $\sum_{n=1}^\infty \dfrac{n-1}{n^2\sqrt{n}}$ converge or diverge?

   (a) Converges since $\dfrac{n-1}{n^2\sqrt{n}} < \dfrac{1}{n^{3/2}}$
   (b) Converges since $\lim_{n \to \infty} \dfrac{\frac{n-1}{n^2\sqrt{n}}}{\frac{1}{n^{3/2}}} = 1.5$
   (c) Diverges since $\dfrac{n-1}{n^2\sqrt{n}} > \dfrac{1}{n}$
   (d) Diverges since $\lim_{n \to \infty} \dfrac{\frac{n-1}{n^2\sqrt{n}}}{\frac{1}{n^{3/2}}} = \infty$

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3. Does $\sum_{n=1}^\infty \dfrac{5n^4 - 3n^3 + 12n^2 + 54n - 78}{27n^5 + 15n^2 + 54n - 78}$ converge or diverge?

   (a) Diverges by limit comparison with $\sum \dfrac{1}{n}$
   (b) Diverges by comparison with $\dfrac{1}{n}$
   (c) Converges by limit comparison with $\sum \dfrac{1}{n^2}$
   (d) Converges by comparison with $\sum \dfrac{1}{n^2}$

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4. Does $\sum_{n=1}^\infty \dfrac{n}{\sqrt{n^5 + 3}}$ converge or diverge?

   (a) Divergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 1$
   (b) Divergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 0$
   (c) Convergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 0$
   (d) Convergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 1$

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5. Does $\sum_{n=1}^\infty \dfrac{8n^{10} - n^6 + 5\sqrt{n}}{n^{12} - n^7 + 2}$ converge or diverge?

   (a) Diverges by limit comparison with $\sum \dfrac{1}{n}$
   (b) Diverges by comparison with $\dfrac{1}{n}$
   (c) Converges by comparison with $\sum \dfrac{1}{n^2}$
   (d) Converges by limit comparison with $\sum \dfrac{1}{n^2}$

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