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
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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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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