## Australopithecus Group Title Use the Maclaurin Series for f(x) using the definition of the Maclaurin Series for sin(pix) Can anyone show me how to find the series using the Maclaurin method? 2 years ago 2 years ago

1. 91 Group Title

f(0) f'(0)(x-0) f''(0)(x-0)^2 ------- + ---------- + ------------- 0! 1! 2!

2. Australopithecus Group Title

I just memorized the table the answer is, but it would be nice to know the method to this as it will probably come up on my final $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi x)^{2n+1}}{(2n+1)!}$

3. 91 Group Title

so let's start taking derivative

4. 91 Group Title

f'(0)= pi cos(pi x) = pi f''(0)=-pi^2 sin(pi x)=0 f'''(0)=-pi^3 cos(pi x)=-pi^3

5. 91 Group Title

so you see the pattern pi, 0, -pi^3,0,pi^5,0,-pi^7

6. Australopithecus Group Title

right

7. 91 Group Title

so you can put it into the series format pi x - pi^3 (x)^3 + pi^5 (x)^5 --------- ----------- - ..... 3! 5!