Ratio and Root Tests Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{(n+2)!}{10^n \cdot n!}$ converge?

   (a) Converges absolutely by the ratio test
   (b) Converges conditionally by the ratio test
   (c) Diverges by the ratio test
   (d) Diverges by the test for divergence

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2. Does $\sum_{n=1}^\infty \dfrac{n^n}{3^n}$ converge or diverge?

   (a) Converges by the ratio test
   (b) Converges by the root test
   (c) Diverges by the $\delta-\varepsilon$ test
   (d) Diverges by the root test

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3. Test $\sum_{n=1}^\infty ne^{-n}$ for convergence.

   (a) Diverges by the ratio test
   (b) Divergent telescoping sum
   (c) Converges since $\int_1^\infty xe^{-x}\;dx = \dfrac{1}{e}$
   (d) Converges by the ratio test

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

   (a) Converges conditionally by the ratio test
   (b) Converges absolutely by the ratio test
   (c) Diverges by the ratio test
   (d) Diverges by the test for divergence

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5. Test $\sum_{n=1}^\infty \dfrac{(-1)^{n-1} n!}{n^n}$ for convergence.

   (a) Convergent binomial distribution
   (b) Converges by the ratio test
   (c) Diverges by the ratio test
   (d) Diverges by the root test

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