Thomas’ Calculus Early Transcendentals 12ed

Homework Assignment for Chapter 10

1. Section 10.1: Problems 53, 59, 63, 67, 69, 81, 87.

2. Section 10.1: Study Definition of the limit of a sequence on page 552. Make sure that you can write down the definition correctly.

3. Section 10.2: Problems 61, 65, 71, 75.

4. Section 10.2: Study the definition of the sum of a series on page 563.

5. Section 10.3: Problems 27, 31, 33, 37, 53, 55.

6. Section 10.4: Problems 17, 27, 29, 31, 43, 45, 51, 61, 62.

Note in problem 62: the case p = 1, q < −1 should be convergent, instead.

7. Section 10.5: Odd numbered problems in problem 17-43, 61, 65.

8. Section 10.6: Problems 4, 11, 25, 26, 28, 29, 30, 35, 39, 43, 49, 53.

9. Section 10.7: 7, 11, 15, 19, 23, 29, 33, 37, 47, 51, 56.

10. Section 10.7: Find a power series that converges on (1, 3) and diverges otherwise. Do the same for (1, 3], [1, 3) and [1, 3], respectively.

11. Section 10.7: Use the power series expression of \frac{1}{1−x} to find that of ln(1−x) on |x| < 1.

12. Section 10.7: Find the first few terms of the power series representation of

\frac{1 − x^{2} + x^{4} − ⋯}{1 − \frac{x^{2}}{2!} + \frac{x^{4}}{4!} − ⋯}

13. Section 10.8: Problems 15, 23, 29, 35.

Remark for problem 23: We know that f(x) = ∑_{n=0}^{3} b_{n}(x − 2)^{n} for some b_{n}’s (for example, one can conclude this by repeated division by (x − 2)). Nevertheless, it is enough to assume f(x) can be written this form. The explicit values of b_{n} is not needed. Show that, the final answer is the same as f(x).

14. Section 10.8: Let

f(x) = \{ \begin{matrix}0, & x = 0 \\ e^{−1/x^{2}}, & x ≠ 0 \end{matrix}

It is known that f^{(n)}(0) = 0 for all n. Verify this for f′(0) and f′′(0).

15. Section 10.9: Problems 7, 9, 17, 19, 33, 41, 42, 50(a).

16. Section 10.10: Problems 10, 27, 35, 37, 43, 46, 51, 64, 66.