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
$$
\frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}}
$$

Adding the terms together, you should get:
$$
\sum_{\text{all terms}} = \frac{(32 x^{31})}{(x^{32}-y^{32})}
$$
This result is obtained by using the LCD, (Least Common Denominator).  
LCD:
$$
(x-y) (x+y) (x^2+y^2) (x^4+y^4) (x^8+y^8) (x^{16}+y^{16}) =\\ (x^2-y^2)(x^2+y^2)(x^4+y^4) (x^8+y^8) (x^{16}+y^{16})=\\ 
(x^4-y^4)(x^4+y^4)(x^8+y^8) (x^{16}+y^{16})=\\
(x^8-y^8)(x^8+y^8)(x^{16}+y^{16})=\\
(x^{16}-y^{16})(x^{16}+y^{16})=\\
(x^{32}-y^{32}) =\\
\text{LCD(LCD(LCD(LCD(LCD( $x-y$, $x+y$), $x^2+y^2$), $x^4+y^4$), $x^8 + y^8$), $x^{16}+y^{16}$)}
$$
Note
It would be preferable to move from left to right simplifying the pairwise additions.
I included the LCD so you could see the general pattern in the denominators as you progress in your simplification from left to right. 
