Summation Notation

Question

How do I write the taylor series for

f(x) = (6x^2) / (1-2x^3)^2 at x=0

in summation notation??

Answer 1

This is a Maclaurin series. Turns out that if you expand the Maclaurin series, only every 3rd term is non-zero.

f(x) = 6x²+24x⁵+72x⁸+ ...

Fitting the coefficients and exponents to a parabola and a line

∞

f(x) = ∑(15n²+3n+6)x³ⁿ⁺² = 6x²+24x⁵+72x⁸+150x¹¹+....

n=0

I plotted both the original function and the Maclaurin function and it is very accurate around zero.

Answer 2

I don't know why you would want to, but simply do the long division to get the first few terms...after that you may be able to "guess" the general term or resort to the finding the derivatives (and good luck with that!)