Alternating Series Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{(-1)^n\sqrt{n}}{n+10}$ converge or diverge?

   (a) Divergent by the test for divergence
   (b) Divergent telescoping series
   (c) Convergent since it is absolutely convergent
   (d) Convergent since $\dfrac{\sqrt{n}}{n+10}$ decreases to $0$

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

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

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3. Does the series $\sum_{n=1}^\infty (-1)^n \sin{\left(\dfrac{1}{n}\right)}$ converge or diverge?

   (a) Divergent by the test for divergence
   (b) Divergent by the integral test
   (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. Determine whether or not $\sum_{n=1}^\infty \dfrac{1 + \cos{\left(2n\right)}}{3^n}$ converges.

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

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