Calcular una aproximación de ln (1?

Hallemos la aproximación hasta el tercer grado por medio de la serie de Taylor en el entorno de x = 0 <img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0Af%28x%29%3D%5Csum_%7Bi%3D0%7D%5E%7B3%7D%5Cdfrac%7Bf%5E%7B%28i%29%7D%280%29%7D%7Bi%21%7Dx%5Ei%2Bo%28x%5E3%29%5C%5C%20%5C%5C%20%5C%5C%0Af%28x%29%3D%20f%280%29%2Bf%27%280%29x%2B%5Cdfrac%7Bf%27%27%280%29%7D%7B2%7Dx%5E2%2B%5Cdfrac%7Bf%27%27%27%280%29%7D%7B6%7Dx%5E3%2Bo%28x%5E3%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BHallemos%20las%20derivadas%20sucesivas%20de%20la%20funci%5C%27on%20%7D%5C%5C%20%0Af%28x%29%3D%5Cln%281%2Bx%29%5C%5C%20%5C%5C%20%5C%5C%0Af%280%29%3D0%5C%5C%20%5C%5C%0Af%27%28x%29%3D%5Cdfrac%7B1%7D%7B1%2Bx%7D%5Cto%20f%27%280%29%3D1%5C%5C%20%5C%5C%0Af%27%27%28x%29%3D-%5Cdfrac%7B1%7D%7B%281%2Bx%29%5E2%7D%5Cto%20f%27%27%280%29%3D-1%5C%5C%20%5C%5C%0Af%27%27%27%28x%29%3D%5Cdfrac%7B2%7D%7B%281%2Bx%29%5E3%7D%5Cto%20f%27%27%280%29%3D2%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BPor%20tanto%3A%20%7D%5C%5C%20%5C%5C%0Af%28x%29%5Csim%20x-%5Cdfrac%7B1%7D%7B2%7Dx%5E2%2B%5Cdfrac%7B1%7D%7B3%7Dx%5E3" />

<img src="https://tex.z-dn.net/?f=%5Cln%281.4%29%3Df%280.4%29%5Csim%200.4-%5Cdfrac%7B1%7D%7B2%7D%280.4%29%5E2%2B%5Cdfrac%7B1%7D%7B3%7D%280.4%29%5E3%3D%5Cdfrac%7B128%7D%7B375%7D%5C%5C%20%5C%5C%20%5C%5C%0A" />.