Ratio and Root Tests Quiz

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1. Test $\sum_{n=1}^\infty \dfrac{(-1)^{n+1}(6+n)5^n}{(n^2)5^{2n}}$ 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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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. Does $\sum_{n=1}^\infty \dfrac{n^n}{n!}$ converge or diverge?

   (a) Converges by the root test
   (b) Converges by the ratio test
   (c) Diverges by the ratio test
   (d) Diverges by the closed interval method

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4. 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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5. 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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