Calculus II, Spring 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 04

Assigned Mar 19, 2009.

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

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

3. Section 10.4: Problems 9, 13, 17, 23, 28, 29.

4. Solve the differential equation

dy/dx = 1 + y^{2}, y(0) = 0,

by power series expansion. The differential equation can also be integrated directly. Verify your answer by comparing the first 2 nonzero coefficients of the Taylor series expansion of the solution, which can be found on p624.

5. Read p 619, which says the Taylor series generated by (1 + x)^{m}, m ∈ R, converges on |x| < 1. It is not shown there that the series indeed converges to (1 + x)^{m}. Analyzing R_{n}(x) directly is not quite easy (try it and youll see why). An alternative approach is through the following steps:

(a) Verify that

(k + 1) ( \begin{matrix}m \\ k + 1 \end{matrix} ) + k ( \begin{matrix}m \\ k \end{matrix} ) = m ( \begin{matrix}m \\ k \end{matrix} )

(b) Define, for |x| < 1,

f(x) = ∑_{k=0}^{∞} ( \begin{matrix}m \\ k \end{matrix} ) x^{k} = 1 + mx + \frac{m(m − 1)}{2!}x^{2} \frac{m(m − 1)(m − 2)}{3!}x^{3} + ⋯

Show that f(0) = 1 and

(1 + x)f′(x) = mf(x), |x| < 1.

(c) Show that f(x) = (1 + x)^{m} on |x| < 1.