\(P=\frac{1}{1-x}+\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{4}{x^4+1}+\frac{8}{x^8+1}+\frac{16}{x^{16}+1}\)
\(P=\frac{x+1+1-x}{1-x^2}+\frac{2}{x^2+1}+\frac{4}{x^4+1}+\frac{8}{x^8+1}+\frac{16}{x^{16}+1}\)
\(P=\left(\frac{2}{1-x^2}+\frac{2}{x^2+1}\right)+\frac{4}{x^4+1}+\frac{8}{x^8+1}+\frac{16}{x^{16}+1}\)
\(P=\left(\frac{4}{1-x^4}+\frac{4}{x^4+1}\right)+\frac{8}{x^8+1}+\frac{16}{x^{16}+1}\)
\(P=\frac{8}{1-x^8}+\frac{8}{x^8+1}+\frac{16}{x^{16}+1}=\frac{8x^8+8+8-8x^8}{\left(1-x^8\right)\left(1+x^8\right)}+\frac{16}{x^{16}+1}\)
\(P=\frac{16}{1-x^{16}}+\frac{16}{x^{16}+1}=\frac{32}{\left(1-x^{16}\right)\left(1+x^{16}\right)}=\frac{32}{1-x^{32}}\)
