User:Egm6321.f10.team6.Kim.MK/hw7

= Show the character of Even and Odd Functions (Problem 7.2)=

From class note lecture 38_1

Problem Statement
Let

Required
Show that: 1. if the $$ \begin{align} \{g_i\} \end{align}$$ is odd, then $$ \begin{align} f \end{align}$$ is odd.

2. if the $$ \begin{align} \{g_i\} \end{align}$$ is even, then $$ \begin{align} f \end{align}$$ is even.

==Solution ==

1. The definition of an odd function is that $$\begin{align} f(x) = -f(-x) \end{align}$$.

If $$\begin{align}\sum_{i=0}^{n} g_i(x)\end{align}$$ is odd function,

 Odd functions, $$\begin{align} y = x^{2n+1}\end{align}$$

$$\begin{align} y= sin (nx) \end{align}$$

2. The definition of an even function is that $$\displaystyle f(x) = f(-x)$$.

If $$\begin{align}\sum_{i=0}^{n} g_i(x)\end{align}$$ is even function,  Even functions, $$\begin{align} y = x^{2n}\end{align}$$

$$\begin{align} y= cos (nx) \end{align}$$

Problem Statement
We have recurrence relations that

Binomial Theorem :

Thus,

Required
Continue power series expansion to find $$\begin{align}{P_3, P_4 , P_5 , P_6 } \end{align}$$, and compare the result to that obtained by a) (7) & (8) class note lecture 36_2

b) the result of HW 7.9

a)
We need the first 7 terms of the expansion

Using (10.5), we get

Therefore, we have confirmed two results are equivalent.

b)
Problem 9 result