Question

利用函数的幂级数展开式求ln3的近似值（误差不超过0.0001）请附带过程。谢谢

Answer 1

ln3=2{[(3-1)/(3+1)]+(1/3)[(3-1)/(3+1)]^3+(1/5)[(3-1)/(3+1)]^5+(1/7)[(3-1)/(3+1)]^7+(1/9)[(3-1)/(3+2)]^9+...+ }.
=2[(1/2)+(1/3)*(1/2)^3+(1/5)(1/2)^5+(1/7)(1/2)^7+(1/9)(1/2)^9+...].
=2(0.50000+0.04166+0.00625+0.00111+0.000217+...)
=2*0.54923.
=1.09846
∴ ln3≈1.0985.

Answer 2

1/(1+x^4)=1-x^4+x^8-.... ∫[0,0.5]1/(1+x^4)dx=∫[0,0.5](1-x^4+x^8)dx+o(x^(13)) ≈1/2-