maclaurin series for function undefined at a point

Question

Say i want a power series for a function such as $$\frac{(2x+2)(x)}{(2x)(3x+1)}$$ at $x=0$. How would one go about this? I have acquired the second, third and fourth terms, but am struggling getting the first term since f(0) is undefined.

Can one just assume that for very small $x%$, i.e $\lambda << 1$ then $ f(\lambda)\approx 1$ hence the series takes the form 1+....

Answer 1

The first term is the limit at the point.

We get $$\frac{(2x+2)(x)}{(2x)(3x+1)}=\frac{x+1}{3x+1}=\frac{1}{3}+\frac{2}{3}\frac{1}{3x+1}=\frac{1}{3}+\frac{2}{3}\sum_{n=0}^{\infty}(-1)^n3^nx^n$$

Answer 2

In this case, you might factor a 2 out of the first factor of the top, and then cancel $2x$ on top and bottom. That gets you a new function $x/(3x+1)$ that equals yours almost everywhere, hence they have the same MacLaurin series. The series for YOUR function at 0 isn't really well-defined, but if it were, it'd have to be the series for mine...