Comparison Tests Quiz

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

   (a) Convergent by comparison with $\sum \left(\dfrac{1}{2}\right)^n$
   (b) Convergent by limit comparison with $\sum \left(\dfrac{2}{3}\right)^n$
   (c) Divergent by comparison with $\sum \left(\dfrac{1}{2}\right)^n$
   (d) Divergent by limit comparison with $\sum \left(\dfrac{2}{3}\right)^n$

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

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

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

   (a) Convergent by comparison with $\sum \dfrac{1}{n^2}$
   (b) Convergent by limit comparison with $\sum \dfrac{1}{n^{3/2}}$
   (c) Divergent by comparison with $\sum \dfrac{1}{n}$
   (d) Divergent by limit comparison with $\sum \dfrac{1}{n^{3/2}}$

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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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