Australopithecus
  • 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?
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
f(0) f'(0)(x-0) f''(0)(x-0)^2 ------- + ---------- + ------------- 0! 1! 2!
Australopithecus
  • Australopithecus
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 \[\sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi x)^{2n+1}}{(2n+1)!}\]
anonymous
  • anonymous
so let's start taking derivative

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anonymous
  • anonymous
f'(0)= pi cos(pi x) = pi f''(0)=-pi^2 sin(pi x)=0 f'''(0)=-pi^3 cos(pi x)=-pi^3
anonymous
  • anonymous
so you see the pattern pi, 0, -pi^3,0,pi^5,0,-pi^7
Australopithecus
  • Australopithecus
right
anonymous
  • anonymous
so you can put it into the series format pi x - pi^3 (x)^3 + pi^5 (x)^5 --------- ----------- - ..... 3! 5!

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