Posted by **W** on Thursday, March 8, 2012 at 8:21pm.

Find a power series, centered @ x=0, for function f(x)=x/(1+2x).

I know this is a maclaurin series, but my work doesn't get the right answer. Can you please show steps? Also,do all power series start with a 1, as in (1+2x+4x^2+...)?

Thanks in advance!

- Calculus -
**Steve**, Friday, March 9, 2012 at 11:03am
No, take a look at the definition of the Maclaurin series. It starts with f(0). very power series starts with the first term of its defined sequence.

f = x/(1+2x)

f' = 1/(1+2x)^2

f^{(2)} = -4/(1+2x)^3

f^{(3)} = 24/(1+2x)^4

f^{(4)} = -192/(1+2x)^5

from then on it's just the power rule:

f(n) = (-2)^(n-1) n! (1+2x)^-(n+1)

so, we have

f(0) + 1/1! f^{(1)}(0) + 1/2! f^{(2)}(0) + ...

= 0 + 1x - 4/2!x^2 + 24/3!x^3 - 192/4!x^4 + ...

= 0 + x - 2x^2 + 4x^3 - 8x^4 + ...

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