$\dfrac{1}{1-x}+\dfrac{1}{x+1}+\dfrac{2}{x^2+1}+.....+\dfrac{16}{x^{16}+1}$
$ = \dfrac{1}{x+1}-\dfrac{1}{x-1} + \dfrac{2}{x^2+1} + ....+\dfrac{16}{x^{16}+1}$
$ = \dfrac{-2}{x^2-1} + \dfrac{2}{x^2+1} + \dfrac{4}{x^4+1} +....+\dfrac{16}{x^{16}+1}$
$ = \dfrac{-4}{x^4-1} + \dfrac{4}{x^4+1} + \dfrac{8}{x^8+1} + \dfrac{16}{x^{16}+1}$
$ = \dfrac{-8}{x^8-1} + \dfrac{8}{x^8+1} + \dfrac{16}{x^{16}+1}$
$ = \dfrac{-16}{x^{16}-1} + \dfrac{16}{x^{16}+1}$
$ = \dfrac{-32}{x^{32}-1}$
