Вправа 201 - 300 » 211

211. Доведи тотожність: a) 1/(1-x) + 1/(1+x) + 2/(1+ x^2 ) + 4/(1+ x^4 ) = 8/(1- x^8 ); (1+x+1-x)/((1-x)(1+x)) + (2+ 〖2x〗^4+ 4+ 〖4x〗^2)/((1+ x^2 )(1+ x^4)) = 8/(1- x^8 ); (〖2(1+x〗^2)(〖1+x〗^4 )+ (〖2x〗^4+ 〖4x〗^2+ 6)(1- x^2))/((1- x^2 )(1+ x^2 )(1+ x^4)) = 8/(1- x^8 ); (2(1+ x^4+ x^2+ x^6 )+ 〖2x〗^4- 〖2x〗^6+ 〖4x〗^2- 〖4x〗^4+ 6- 〖6x〗^2)/((1- x^2 )(1+ x^2 )(1+ x^4)) = 8/(1- x^8 ); (2+ 〖2x〗^4+ 〖2x〗^2+ 〖2x〗^6+ 〖2x〗^4- 〖2x〗^6+ 〖4x〗^2- 〖4x〗^4+ 6- 〖6x〗^2)/((1- x^2 )(1+ x^2 )(1+ x^4)) = 8/(1- x^8 ); 8/(1- x^8 ) = 8/(1- x^8 ). Тотожність доведено. б) 1/(1-2x) + 1/(1+2x) + 2/(1+4x^4 ) + 4/(1+16x^4 ) = 8/(1-256x^8 ); (1+2x+1-2x)/(1-4x^2 ) + 2/(1+4x^2 ) + 4/(1+16x^4 ) = 8/(1-256x^8 ); (2(1+4x^2 )+ 25(1-4x^2))/((1-4x^2 )(1+4x^2)) + 4/(1+16x^4 ) = 8/(1-256x^8 ); (2+8x^2+ 2-8x^2)/(1-256x^8 ) = 8/(1-256x^8 ); 8/(1-256x^8 ) = 8/(1-256x^8 ). Тотожність доведено.