Also schon das Taylorpolynom ist nicht wirklich richtig
a)
f(x) = (1 + x)^{2/3}
f'(x) = 2/(3·(x + 1)^{1/3})
f''(x) = - 2/(9·(x + 1)^{4/3})
f'''(x) = 8/(27·(x + 1)^{7/3})
f''''(x) = - 56/(81·(x + 1)^{10/3})
f'''''(x) = 560/(243·(x + 1)^{13/3})
T4(x) = f(0) + f'(0)/1!·x^1 + f''(0)/2!·x^2 + f'''(0)/3!·x^3 + f''''(0)/4!·x^4
T4(x) = 1 + 2/3·x^1 - 1/9·x^2 + 4/81·x^3 - 7/243·x^4
T4(1/3) = 1 + 2/3·(1/3)^1 - 1/9·(1/3)^2 + 4/81·(1/3)^3 - 7/243·(1/3)^4 = 23843/19683 = 1.211349895
Vergleich zum realen Wert
(4/3)^{2/3} = 1.211413728