Alternating Series Quiz

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1. Does $\sum_{n=1}^\infty (-1)^n\dfrac{\sqrt{n}}{n+3}$ converge?

   (a) Divergent by the test for divergence
   (b) Divergent telescoping series
   (c) Convergent since it is absolutely convergent
   (d) Convergent since $\dfrac{\sqrt{n}}{n+3}$ decreases to $0$

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

   (a) Convergent telescoping series
   (b) Convergent since $\dfrac{n}{\ln{n}}$ decreases to $0$
   (c) Divergent by the test for divergence
   (d) Divergent telescoping series

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3. Does $\sum_{n=1}^\infty \dfrac{\cos{\left(n\pi\right)}}{\ln{\left(n\right)}}$ converge?

   (a) Divergent by the test for divergence
   (b) Divergent by comparison with a $p$-series
   (c) Convergent by the alternating series test (but not absolutely convergent)
   (d) Absolutely convergent

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4. Test $\sum_{n=1}^\infty \dfrac{(-1)^{n-1}5^{n-1}}{5^{n+1}n^{5/6}}$ for convergence.

   (a) Diverges by the rational root theorem
   (b) Diverges by the test for divergence
   (c) Converges since it is absolutely convergent
   (d) Converges since $\dfrac{1}{25n^{5/6}}$ decreases to 0

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5. Does the series $\sum_{n=1}^\infty \dfrac{\cos{\left(\frac{n}{e^{\pi/e^2}}\right)}}{n^2}$ converge or diverge?

   (a) Conditionally convergent
   (b) Absolutely convergent
   (c) Divergent by the test for divergence
   (d) Divergent by the integral test

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