Alternating Series Quiz

Does the series $\sum_{n=5}^\infty (-1)^{n-1}\dfrac{1}{\sqrt{n^3 - 5}}$ converge? If so, does it converge absolutely?

(a) Absolutely convergent by limit comparison with $\sum \dfrac{1}{n^{3/2}}$

(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

Does $\sum_{n=5000}^\infty \dfrac{(-1)^n}{2n + 3}$ converge? If so, does it converge absolutely?

(a) Divergent by the test for divergence

(b) Convergent (but not absolutely convergent by comparison with $\sum \frac{1}{n}$)

(c) Convergent (but not absolutely convergent by limit comparison with $\sum \frac{1}{n}$)

(d) Absolutely convergent by limit comparison with $\sum \dfrac{1}{n^2}$

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

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

Is $\sum_{n=1}^\infty \dfrac{(-1)^{n-1}(n^3+1)}{n^4+2}$ convergent? Absolutely convergent?

(a) Divergent by the test for divergence

(b) Convergent (but not absolutely convergent by comparison with $\sum \frac{1}{n}$)

(c) Convergent (but not absolutely convergent by limit comparison with $\sum \frac{1}{n}$)

(d) Absolutely convergent by limit comparison with $\sum \dfrac{1}{n^2}$

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