Ratio and Root Tests Quiz

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

   (a) Converges conditionally by the ratio test
   (b) Diverges by the ratio test
   (c) Converges absolutely by the root test
   (d) Converges conditionally 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 $\sum_{n=1}^\infty \dfrac{2^n \cdot n!}{n^n}$ 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. 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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5. Does $\sum_{n=1}^\infty (1 + 1/n)^{n^2}$ converge or diverge?

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

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