Megoldani a matematikai problémákat

Lépésről lépésre

1. f(x) = \log{\left(x \right)}

   Taylor series about x = 1: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

2. f^{(0)}(1) = \log{\left(x \right)}\big|_{x=1} = 0

   Derivative 0 at the centre.

3. f^{(1)}(1) = \frac{1}{x}\big|_{x=1} = 1

   Derivative 1 at the centre.

4. f^{(2)}(1) = - \frac{1}{x^{2}}\big|_{x=1} = -1

   Derivative 2 at the centre.

5. f^{(3)}(1) = \frac{2}{x^{3}}\big|_{x=1} = 2

   Derivative 3 at the centre.

6. f^{(4)}(1) = - \frac{6}{x^{4}}\big|_{x=1} = -6

   Derivative 4 at the centre.

7. f^{(5)}(1) = \frac{24}{x^{5}}\big|_{x=1} = 24

   Derivative 5 at the centre.

8. \frac{x^{5}}{5} - \frac{5 x^{4}}{4} + \frac{10 x^{3}}{3} - 5 x^{2} + 5 x - \frac{137}{60}

   Assemble the terms up to degree 5.