Alternating Series Quiz

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

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

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

   (a) Convergent since the series is absolutely convergent
   (b) Convergent since $\dfrac{1}{n\ln{n}}$ 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 $\sum_{n=1}^\infty \dfrac{(-1)^n\sqrt{n}}{1 + 2\sqrt{n}}$ converge or diverge?

   (a) Absolutely convergent by limit comparison with $\sum \dfrac{1}{n^{1/2}}$
   (b) Convergent (but not absolutely convergent by limit comparison with $\sum \frac{1}{n}$)
   (c) Convergent (but not absolutely convergent by comparison with $\sum \frac{1}{n}$)
   (d) Divergent by the test for divergence

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5. Does $\sum_{n=1}^\infty \dfrac{(-1)^n\cos{\left(n\right)}}{n^2}$ converge?

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

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