Comparison Tests Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{(2n+3)}{(n+2)^2}$ converge?

   (a) Converges since $\lim_{n \to \infty} \dfrac{n(2n+3)}{(n+2)^2} = 2$
   (b) Converges since $\dfrac{2n+3}{(n+2)^2} < \dfrac{1}{n}$
   (c) Diverges since $\lim_{n \to \infty} \dfrac{n(2n+3)}{(n+2)^2} = 2$
   (d) Diverges since $\dfrac{2n+3}{(n+2)^2} < \dfrac{1}{n}$

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2. Does $\sum_{n=1}^\infty \dfrac{5n^5}{6n^6}$ converge?

   (a) Diverges since $\dfrac{5n^5}{6n^6} > \dfrac{1}{n}$
   (b) Converges since $\dfrac{5n^5}{6n^6} < \dfrac{1}{n^2}$
   (c) Converges since this is a $p$-series with $p > 1$
   (d) Diverges since this is a $p$-series with $p \leq 1$

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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 $\sum_{n=1}^\infty \dfrac{1}{n + \sqrt{n}}$ 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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5. Does $\sum_{n=1}^\infty \dfrac{1}{n!}$ converge?

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

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