The question requires me to obtain the binomial expansion of $$\sqrt[4]{1+x}+\sqrt[4]{1-x} $$ up to term containing $x^2$.
My initial attempt involved finding the binomial expansion of each side up to the term with $x^2$ which resulted in:
$$ 1+\frac{x}{4}-\frac{3}{32}x^2+\dots $$ and
$$ 1-\frac{x}{4}-\frac{3}{32}x^2+\dots $$ and then I thought of simply adding the expansions but then I wondered if it was possible since each side is infinite. Is it okay to add both sides since I'm only finding terms up to the one containing $x^2$?
Yes, you can add them because the remainder in both cases is $o(x^2)$ (see little-o notation): $$\sqrt[4]{1\pm x}=1\pm \frac{x}{4}-\frac{3}{32}x^2+o(x^2).$$ Hence $$\sqrt[4]{1+x}+\sqrt[4]{1-x}=2-\frac{3}{16}x^2+o(x^2).$$