Alternating Series Quiz

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

   (a) Divergent by the test for divergence
   (b) Divergent telescoping series
   (c) Convergent since it is absolutely convergent
   (d) Convergent since $\dfrac{\sqrt{n^2+1}}{n^2}$ decreases to $0$

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2. Is $\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{n^2 + 5}$ convergent?

   (a) Convergent telescoping series
   (b) Convergent since $\dfrac{1}{n^2+5}$ 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. Does the series $\sum_{n=5}^\infty (-1)^{n-1}\dfrac{1}{\sqrt{n^2 - 5}}$ converge? If so, does it converge absolutely?

   (a) Convergent since $\dfrac{1}{\sqrt{n^2-5}}$ decreases to $0$ (but not absolutely convergent by limit comparison with $\sum \frac{1}{n}$)
   (b) Convergent since it is absolutely convergent by comparison with $\sum \dfrac{1}{n^2}$
   (c) Divergent telescoping series
   (d) Divergent by the test for divergence

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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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