Calculus II, Spring 2016

Brief answers to Quiz 1

Mar 03, 2016 Show all details.

1. (20 pts) Give formal definition of lim_{n→∞} a_{n} = L

Ans. See page 552 of the textbook. If you did not get full credit, make sure you realize what was the mistake in your answer.

2. (16 pts) Is ∫0^{\frac{π}{2}} \sqrt{tan t} dt convergent? Explain.

Ans. ∵ lim_{t→π/2−} \frac{\sqrt{tan t}}{\frac{1}{\sqrt{\frac{π}{2} −t}}} = 1. (6 pts) And, ∫_{0}^{π/2} _{\frac{1}{\sqrt{\frac{π}{2} −t}}} dt = ∫_{0}^{π/2} \frac{1}{\sqrt{u}} du converges. (6 pts)

∴ By Limit Comparison Test, ∫0^{\frac{π}{2}} \sqrt{tan t} dt converges. (4 pts)

3. (16 pts) Is ∑_{n=1}^{∞} (1 + \frac{1}{n})^{n} convergent? Explain.

Ans. ∵ (1 + \frac{1}{n})^{n} → e (12 pts). ∴ By Nth Term Test, ∑_{n=1}^{∞}(1 + \frac{1}{n})^{n} divergent (4 pts).

4. (16 pts) Is ∑_{n=3}^{∞} \frac{1}{n(ln n)^{2}} convergent? Explain.

Ans. Let f(x) = \frac{1}{x(ln x)^{2}} (3 pts). Then f is positive, continuous, and decreasing on [3, ∞) (3 pts). ∵ ∫_{3}^{∞} \frac{1}{x(ln x)^{2}} dx = ∫_{ln 3}^{∞} \frac{1}{u^{2}} du converges. (6 pts) ∴ By Integral Test, ∑_{n=3}^{∞} \frac{1}{n(ln n)^{2}} converges (4 pts).

5. (16 pts) Is ∑_{n=1}^{∞} \frac{1}{\sqrt{n^{3} + 1}} convergent? Explain.

∵ lim_{n→∞} \frac{\frac{1}{\sqrt{n^{3}+1}}}{\frac{1}{n \frac{3}{2}}} = 1 (6 pts). And, ∑_{n=1}^{∞} \frac{1}{n^{\frac{3}{2}}} converges (6 pts).

∴ By Limit Comparison Test, ∑_{n=1}^{∞} \frac{1}{\sqrt{n^{3}+1}} converges (4 pts).

6. (16 pts) Is ∑_{n=1}^{∞} \frac{2^{n}n!n!}{(2n)!} convergent? Explain.

Ans. Let a_{n} = \frac{2^{n}n!n!}{(2n)!}.

∵ lim_{n→∞} \frac{a_{n+1}}{a_{n}} = lim_{n→∞} \frac{2(n+1)^{2}}{(2n+1)(2n+2)} = \frac{1}{2} (12 pts). ∴ By Ratio Test, ∑_{n=1}^{∞} \frac{2^{n}n!n!}{(2n)!} converges (4 pts).