Question

用简便方法计算:1-x分之1+1+x分之1+1+x方分之2+1+x的4次方分之4-1-x的8次方分之8

Answer 1

原式=1/(1-x)+1/(1+x)+2/(1+x²)+4/(1+x^4)-8/(1-x^8)

=(1+x+1-x)/[(1-x)(1+x)]+2/(1+x²)+4/(1+x^4)-8/(1-x^8)

=2/(1-x²)+2/(1+x²)+4/(1+x^4)-8/(1-x^8)

=[2(1+x²)+2(1-x²)]/[(1-x²)(1+x²)]+4/(1+x^4)-8/(1-x^8)

=4/(1-x^4)+4/(1+x^4)-8/(1-x^8)

=[4(1+x^4)+4(1-x^4)]/[(1+x^4)(1-x^4)]-8/(1-x^8)

=8/(1-x^8)-8/(1-x^8)

=0