Question

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?

Answer 1

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

Answer 2

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

\[\sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi x)^{2n+1}}{(2n+1)!}\]

Answer 3

so let's start taking derivative

Answer 4

f'(0)= pi cos(pi x) = pi

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

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

Answer 5

so you see the pattern

pi, 0, -pi^3,0,pi^5,0,-pi^7

Answer 6

right

Answer 7

so you can put it into the series format

pi x - pi^3 (x)^3 + pi^5 (x)^5
--------- ----------- - .....
3! 5!