Calculus II, Spring 2013 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Chap 09

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

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

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

4. Section 9.4: Problems 3, 5, 7, 9, 11, 13, 15, 19, 23, 25, 28, 34, 35, 39, 40, 41, 44 (Does it converge? Why?).

5. Section 9.5: Problems 4, 8, 7, 9, 24, 25, 33, 35, 37, 43, 45, 47.

6. We know that if ∑_{n=1}^{∞}a_{n} converges, then lim_{n→∞}a_{n} = 0. How about the counter part for integration? Suppose that f(x) is non-negative, continuous on [0, ∞) and ∫_{0}^{∞}f(x) dx < ∞. Is lim_{x→∞}f(x) = 0 necessarily true?

7. Is ∑_{n=1}^{∞} sin \frac{1}{n} convergent? How about ∑_{n=1}^{∞}(1 − cos \frac{1}{n})?

8. We know that the p = 1/2 series ∑_{k=1}^{∞} k^{\frac{−1}{2}} diverges. The question here is how fast does

the partial sum grows with n, or ∑_{k=1}^{n} k^{\frac{−1}{2}} = O(n^{?})? In other words, can you evaluate

lim_{n→∞} \frac{log (∑_{k=1}^{n}k^{\frac{−1}{2}} )}{log n} if it exists?

9. Section 9.6: Problems 11, 15, 21, 23, 27, 28, 35, 39, 41, 44, 47.

10. Section 9.7: Problems 1, 3, 7, 15, 19, 25, 29, 33 (show that equality holds), 35, 47(a), 50, 57, 58.

11. Find a power series whose interval of convergence is [1, 3). Do the same for (1, 3), [1, 3] and (1, 3], respectively.

12. Find the interval of convergence for the power series

1 + \frac{1}{2}x + \frac{1 · 3}{2 · 4}x^{2} + \frac{1 · 3 · 5}{2 · 4 · 6}x^{3} + ⋯ + \frac{1 ⋯ (2n − 1)}{2 ⋯ 2n}x^{n} + · · · .

Hint: Denote by a_{n} = \frac{1···(2n−1)}{2···2n} and define b_{2} = \frac{2}{3}, b_{3} = \frac{2·4}{3·5}, ⋯, b_{n} = \frac{2···(2n−2)}{3···(2n−1)}. For x = 1, the fact that b_{n} < 1 will help. For x = −1, compare a_{n} with b_{n} and 2b_{n} to conclude that a_{n} ∼ n^{−p}. What is p? Do the same for

1 + \frac{1}{3}x + \frac{1 · 4}{3 · 6}x^{2} + \frac{1 · 4 · 7}{3 · 6 · 9}x^{3} + ⋯ + \frac{1 ⋯ (3n − 2)}{3 ⋯ 3n}x^{n} + · · · .

13. Section 9.8: Problems 9, 13, 28, 29, 31, 32, 33, 34(a), 35, 43.

14. Chap 9: Problems 27, 28, 29, 30, 32, 37, 47(a), 49-54(important), 71, 72.