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Question Number 185449 by Mastermind last updated on 22/Jan/23

$$\mathrm{Find}\mathrm{the}\mathrm{first}\mathrm{four}\mathrm{terms}\mathrm{in}\mathrm{the}$$$$\mathrm{expansion}\mathrm{of}\frac{\mathrm{x}-3}{{(1-{\mathrm{x}}^2)}^2(2+{\mathrm{x}}^2)}\mathrm{in}$$$$\mathrm{ascending}\mathrm{power}\mathrm{of}\mathrm{x}.$$$$$$$$$$$$\mathrm{M}.\mathrm{m}$$

Answered by mr W last updated on 22/Jan/23

$$-\frac{1}{2}(3-x)\underset{k=0}{\sum ^{\mathrm{\infty }}}{C}_1^{k+1}{x}^{2k}\underset{r=0}{\sum ^{\mathrm{\infty }}}{(-1)}^r\frac{x^{2r}}{2^r}$$$$x^{0}term:$$$$-\frac{3}{2}\times 1\times 1=-\frac{3}{2}✓$$$$x^{1}term:$$$$-\frac{1}{2}\times (-1)\times 1\times 1=\frac{1}{2}✓$$$$x^{2}term:$$$$-\frac{3}{2}\times \{2\times 1+1\times (-\frac{1}{2})\}=-\frac{9}{4}✓$$$$x^{3}term:$$$$-\frac{1}{2}\times (-1)\times \{2\times 1+1\times (-\frac{1}{2})\}=\frac{3}{4}✓$$$$....$$

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