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 $\sum_{n=6}^\infty \dfrac{\sqrt{n}}{n-5}$ converge?

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

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

   (a) Convergent since $\dfrac{\sqrt{n^2 + 1}}{n^2} < \dfrac{1}{n^2}$
   (b) Convergent since $\dfrac{\sqrt{n^2+1}}{n^2} > \dfrac{1}{n}$
   (c) Divergent since $\dfrac{\sqrt{n^2+1}}{n^2} > \dfrac{1}{n}$
   (d) Divergent since $\dfrac{\sqrt{n^2+1}}{n^2} < \dfrac{1}{n^2}$

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

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

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