Ratio and Root Tests Quiz

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1. Does $\sum_{n=1}^\infty (-1)^n \dfrac{n}{\ln{(n)}^n}$ converge absolutely, converge conditionally, or diverge?

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

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2. Test $\sum_{n=18}^\infty (-1)^{n+1}\left(\dfrac{\ln{\left(n\right)}}{n}\right)^n$ for convergence.

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

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

   (a) Convergent Stirling number of the second kind
   (b) Converges by L'Hospital's rule
   (c) Divergent by the test for divergence
   (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 $\sum_{n=1}^\infty \dfrac{(n!)^3}{(3n)!}$ for convergence.

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

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