Comparison Tests Quiz

Test $\sum_{n=1}^\infty \dfrac{2^n}{1 + 3^n}$ for convergence.

(a) Convergent by comparison with $\sum \left(\dfrac{1}{2}\right)^n$

(b) Convergent by limit comparison with $\sum \left(\dfrac{2}{3}\right)^n$

(c) Divergent by comparison with $\sum \left(\dfrac{1}{2}\right)^n$

(d) Divergent by limit comparison with $\sum \left(\dfrac{2}{3}\right)^n$

Does $\sum_{n=1}^\infty \dfrac{n+7}{n^2}$ converge?

(a) Diverges since $\dfrac{n+7}{n^2} > \dfrac{1}{n^2}$

(b) Diverges since $\dfrac{n+7}{n^2} > \dfrac{1}{n}$

(c) Converges since $\dfrac{n+7}{n^2} < \dfrac{1}{n^2}$

(d) Converges since $\dfrac{n+7}{n^2} < \dfrac{1}{n}$

Does $\sum_{n=1}^\infty \dfrac{1}{n5^n}$ converge or diverge?

(a) Diverges by comparison to $\sum \dfrac{1}{5^n}$

(b) Diverges by comparison to $\sum \dfrac{1}{10^n}$

(c) Converges by comparison to $\sum \dfrac{1}{10^n}$

(d) Converges by comparison to $\sum \dfrac{1}{5^n}$

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

(a) Convergent since $\dfrac{\sqrt{n^2 + 1}}{n^2} < \dfrac{1}{n^2}$

(b) Convergent since $\dfrac{\sqrt{n^2+1}}{n^2} > \dfrac{1}{n}$

(c) Divergent since $\dfrac{\sqrt{n^2+1}}{n^2} > \dfrac{1}{n}$

(d) Divergent since $\dfrac{\sqrt{n^2+1}}{n^2} < \dfrac{1}{n^2}$

Does $\sum_{n=1}^\infty \dfrac{2}{2^n + 5}$ converge?

(a) Diverges by limit comparison with $\sum \dfrac{1}{n}$

(b) Diverges by comparison with $\dfrac{1}{n}$

(c) Converges by comparison with $\sum \dfrac{1}{2^n}$

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

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