Calculus II, Spring 2019 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 04

1. Section 10.7: 29, 33, 37, 47, 51, 56.

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

3. 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!} − ⋯}

4. 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).

5. 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. We have shown this in class for f′(0) and f′′(0). Continue to verify it for f′′′(0).