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Australopithecus
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?
f(0) f'(0)(x-0) f''(0)(x-0)^2 ------- + ---------- + ------------- 0! 1! 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 \[\sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi x)^{2n+1}}{(2n+1)!}\]
so let's start taking derivative
f'(0)= pi cos(pi x) = pi f''(0)=-pi^2 sin(pi x)=0 f'''(0)=-pi^3 cos(pi x)=-pi^3
so you see the pattern pi, 0, -pi^3,0,pi^5,0,-pi^7
so you can put it into the series format pi x - pi^3 (x)^3 + pi^5 (x)^5 --------- ----------- - ..... 3! 5!