Alternating Series Quiz

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1. Test $\sum_{n=1}^\infty \dfrac{(-1)^n\ln{\left(n+6\right)}}{n^2}$ for convergence.

   (a) Convergent since it is absolutely convergent by comparison to $\sum \dfrac{1}{n^{3/2}}$
   (b) Convergent (but not absolutely convergent by limit comparison to $\sum \frac{1}{n}$)
   (c) Divergent by the test for divergence
   (d) Convergent by the Monotone Convergence Theorem

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

   (a) Convergent telescoping series
   (b) Convergent since $\cos{\left(\frac{1}{n}\right)}$ decreases to $0$
   (c) Divergent by the test for divergence
   (d) Divergent telescoping series

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3. Does the series $\sum_{n=1}^\infty \dfrac{\cos{\left(\frac{n}{e^{\pi/e^2}}\right)}}{n^2}$ 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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4. 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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5. Is $\sum_{n=1}^\infty \dfrac{\sin^2{n}}{n\sqrt{n}}$ absolutely convergent, conditionally convergent, or divergent?

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

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