Power series development for $\frac{1}{4-x^4}$

Question

I have the following problem where I am to find a development using power series for:

$$ \frac{1}{4-x^4} $$

But I have not grasp the concept of such so I would like assistance in understanding the concept and developing the solution for the given problem.

Answer 1

Hint:

Factor out $4$: $$\smash[t]{\frac{1}{4-x^4}=\frac14\frac{1}{1-\smash[b]{\cfrac{x^4}4}}},$$ and set $t=\dfrac{x^4}4$.

Answer 2

Assume that you have been able to write

$$f(x)=a_0+a_1x+a_2x^2+\ldots$$

Then clearly $a_0=f(0)$. So this already tells you what $a_0$ has to be if there is any hope for such a development: $$a_0=\dfrac{1}{4}$$

Now to get access to $a_1$, consider the derivative of the above expression:

$$f^\prime(x)=a_1+2a_2x+\ldots$$ and iterate :)

Now this is what you would do if the function $f$ was a wild one for which nothing is known yet. But for simple functions like yours, everything is known. The first thing you learn is $$\dfrac{1}{1-x}=1+x+x^2+\ldots$$ Via a change of variable you can use this result - as was indicated by Bernard.