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قدم ب قدم

1. f(x) = \frac{1}{- x^{2} - x + 1}

   Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

2. f^{(0)}(0) = \frac{1}{- x^{2} - x + 1}\big|_{x=0} = 1

   Derivative 0 at the centre.

3. f^{(1)}(0) = \frac{2 x + 1}{\left(- x^{2} - x + 1\right)^{2}}\big|_{x=0} = 1

   Derivative 1 at the centre.

4. f^{(2)}(0) = \frac{2 \left(- \frac{\left(2 x + 1\right)^{2}}{x^{2} + x - 1} + 1\right)}{\left(x^{2} + x - 1\right)^{2}}\big|_{x=0} = 4

   Derivative 2 at the centre.

5. f^{(3)}(0) = \frac{6 \left(2 x + 1\right) \left(\frac{\left(2 x + 1\right)^{2}}{x^{2} + x - 1} - 2\right)}{\left(x^{2} + x - 1\right)^{3}}\big|_{x=0} = 18

   Derivative 3 at the centre.

6. f^{(4)}(0) = \frac{24 \left(- \frac{\left(2 x + 1\right)^{4}}{\left(x^{2} + x - 1\right)^{2}} + \frac{3 \left(2 x + 1\right)^{2}}{x^{2} + x - 1} - 1\right)}{\left(x^{2} + x - 1\right)^{3}}\big|_{x=0} = 120

   Derivative 4 at the centre.

7. f^{(5)}(0) = \frac{120 \left(2 x + 1\right) \left(\frac{\left(2 x + 1\right)^{4}}{\left(x^{2} + x - 1\right)^{2}} - \frac{4 \left(2 x + 1\right)^{2}}{x^{2} + x - 1} + 3\right)}{\left(x^{2} + x - 1\right)^{4}}\big|_{x=0} = 960

   Derivative 5 at the centre.

8. 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1

   Assemble the terms up to degree 5.