Comparison Tests Quiz

Does $\sum_{n=1}^\infty \dfrac{n}{\sqrt{n^5 + 3}}$ converge or diverge?

(a) Divergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 1$

(b) Divergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 0$

(c) Convergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 0$

(d) Convergent since $\lim_{n \to \infty} \dfrac{n^{5/2}}{\sqrt{n^5+3}} = 1$

Does $\sum_{n=1}^\infty \dfrac{n+3}{n^{3/2}}$ converge or diverge?

(a) Diverges since $\dfrac{n+3}{n^{3/2}} > \dfrac{1}{\sqrt{n}}$

(b) Diverges since $\dfrac{n+3}{n^{3/2}} > \dfrac{1}{n^2}$

(c) Converges since $\dfrac{n+3}{\sqrt{n}} < \dfrac{1}{n^2}$

(d) Converges since $\dfrac{n+3}{\sqrt{n}} < \dfrac{1}{n}$

Does $\sum_{n=1}^\infty \dfrac{\sin^2{(3n)}}{n^2}$ converge or diverge?

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

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

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

(d) Diverges by the test for divergence

Does $\sum_{n=1}^\infty \dfrac{n}{\sqrt{n^7 + 3}}$ converge or diverge?

(a) Convergent since $\lim_{n \to \infty} \dfrac{n^{7/2}}{\sqrt{n^7 + 3}} = 5/2$

(b) Convergent since $\lim_{n \to \infty} \dfrac{n^{7/2}}{\sqrt{n^7 + 3}} = 1$

(c) Divergent since $\lim_{n \to \infty} \dfrac{n^{7/2}}{\sqrt{n^7 + 3}} = \infty$

(d) Divergent since $\lim_{n \to \infty} \dfrac{n^{7/2}}{\sqrt{n^7 + 3}} = 1$

Does $\sum_{n=1}^\infty \dfrac{1}{n!}$ converge?

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

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

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

(d) Divergent by comparison with $\sum \dfrac{1}{n}$

Infinite Series Main Page