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. Does $\sum_{n=1}^\infty \left(\dfrac{n}{2n+1}\right)^n$ converge 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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3. Does $\sum_{n=1}^\infty \dfrac{27}{(5 + 1/n)^{2n}}$ converge or diverge?

   (a) Converges by the root test
   (b) Converges by Stirling's approximation
   (c) Diverges by the ratio test
   (d) Diverges by the root test

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4. Does $\sum_{n=1}^\infty \dfrac{n^2}{5^n}$ 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 the test for divergence

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

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

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