Simplify $ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $ Please help me find the sum
$$
\frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}}
$$

They're arranged rather nicely. Proceed from left to right and keep using the formula $(a-b)(a+b)=a^2-b^2$ to rewrite the two terms you're about to add with a common (unfactored) denominator. You'll get a good deal of additive cancellation in the numerators, so it won't be all that messy at any stage.
