Ratio and Root Tests Quiz

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1. Test $\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{3^n(n-1)!}$ for convergence.

   (a) Converges by the fundamental theorem of algebra
   (b) Converges by the ratio test
   (c) Diverges by the root test
   (d) Diverges by the test for divergence

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

   (a) Converges absolutely by the ratio test
   (b) Converges conditionally by the ratio test
   (c) Diverges by the ratio test
   (d) Diverges by Dwork's lemma

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

   (a) Diverges by the ratio test
   (b) Diverges by the root test
   (c) Converges by the integral test
   (d) Converges by the ratio test

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4. Test $\sum_{n=1}^\infty \dfrac{(n+1)(7^2 - 1)^n}{7^{2n}}$ 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 $\sum_{n=1}^\infty \dfrac{(n!)^2}{(2n)!}$ for convergence.

   (a) Diverges by the ratio test
   (b) Diverges by the root test
   (c) Converges by the ratio test
   (d) Converges by the factorial test

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