Find-the-first-four-terms-in-the-expansion-of-x-3-1-x-2-2-2-x-2-in-ascending-power-of-x-M-m-

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

Find the first four terms in the

expansion of \frac{x - 3}{{(1 - x^{2})}^{2} (2 + x^{2})} in

ascending power of x .

M . m

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

- \frac{1}{2} (3 - x) \underset{k = 0}{\sum^{\infty}} C_{1}^{k + 1} x^{2 k} \underset{r = 0}{\sum^{\infty}} {(- 1)}^{r} \frac{x^{2 r}}{2^{r}}

x^{0} t e r m :

- \frac{3}{2} \times 1 \times 1 = - \frac{3}{2} ✓

x^{1} t e r m :

- \frac{1}{2} \times (- 1) \times 1 \times 1 = \frac{1}{2} ✓

x^{2} t e r m :

- \frac{3}{2} \times {2 \times 1 + 1 \times (- \frac{1}{2})} = - \frac{9}{4} ✓

x^{3} t e r m :

- \frac{1}{2} \times (- 1) \times {2 \times 1 + 1 \times (- \frac{1}{2})} = \frac{3}{4} ✓

\dots .

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