Maclaurin series for functions which are not infinitely differentiable

Solution 1:

Maclaurin's formula is simply a tool to help you find a power series representation of a function centered at zero, ie: $$f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n + \cdots,$$ where $a_n = f^{(n)}(0)/n!$. Your function is already a power series, though, it's just that only three terms are non-zero:

$$1-12x^2 +\frac{1}{7}x^4 = 1 + 0x - 12 x^2 + 0x^3 + \frac{1}{7}{x^4} + 0x^5 + 0x^6 +\cdots$$

All the remaining terms are zero, so there's not much to do. As you even observed yourself, though, $f^{(n)}(x)=0$ for large enough $n$, so Maclaurin's formula yields the same result.

Solution 2:

I'm sorry to answer my own question. But I was so wrong. I found out you can do it using wolfram alpha (chronic laziness, sorry for that):

http://www.wolframalpha.com/input/?i=maclaurin+expansion+of+17x^4%E2%88%9212x^2%2B1

Spot on...