Comparison Tests Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{1}{n + \sqrt{n}}$ converge?

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

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2. Does $\sum_{n=1}^\infty \dfrac{\ln{\left(5n\right)}}{n}$ converge?

   (a) Diverges since $\dfrac{\ln{(5n)}}{n} > \dfrac{1}{n^2}$
   (b) Diverges since $\dfrac{\ln{(5n)}}{n} > \dfrac{1}{n}$
   (c) Converges since $\dfrac{\ln{(5n)}}{n} < \dfrac{1}{n^2}$
   (d) Converges since $\dfrac{\ln{(5n)}}{n} < \dfrac{1}{n}$

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3. Test $\sum_{n=1}^\infty \dfrac{2^n}{1 + \dfrac{3^n}{1 + 4^n}}$ for convergence.

   (a) Diverges by comparison to $\sum 2^n$
   (b) Diverges by the test for divergence
   (c) Converges by comparison to $\sum \dfrac{1}{2^n}$
   (d) Converges by limit comparison to $\sum \dfrac{1}{2^n}$

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