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)^(n1) 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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