Calculus II, Spring 2012 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 01

1. (s8.6-extra1)

Check if the improper integrals ∫_{0}^{1} x^{−p}dx and ∫_{1}^{∞} x^{−p}dx converge for the cases p > 1, p = 1 and 0 < p < 1, respectively by direct evaluation. Then memorize the results.

2. Section 8.6: Problems: 7, 13, 21, 25, 31, 33, 34, 37, 38, 41, 51, 54, 55, 56, 58, 60, 64, 71, 73. You may find it convenient to use Theorem 1 and Theorem 2 for almost all of the problems. Simply compare the integrand with one of the cases in Problem 1 above.

3. (s8.6-extra2)

For what values of p > 0 is ∫0^{\frac{π}{2}} e^{x} tan^{p} x dx convergent?

4. Section 9.1: Problems: 7, 41, 57, 59, 60, 65, 71, 72(abe).

5. Section 9.2: Problems: 14, 20, 33, 62, 63.

6. Section 9.3: Problems: 23, 24, 27, 39, 41, 47, 61, 62.

7. (s9.3-extra1)

Is ∑_{n=1}^{∞} sin \frac{1}{n} convergent? How about ∑_{n=1}^{∞}(1 − cos \frac{1}{n})? Hint: Try to compare it with ∑ \frac{1}{n^{p}}. The key is to find the correct p first.