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. 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 $\sum_{n=1}^\infty (-1)^n \dfrac{n!}{n^n}$ converge or diverge? You may assume that $n! < (n/2)^n$ for $n \geq 6$.

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

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4. 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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5. 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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