decompose inside R[x] F(x)= (x^(2n) /((x^2 +1)^n )) with n from N and n>0.


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Question Number 30425 by abdo imad last updated on 22/Feb/18

$d e c o m p o s e i n s i d e R [x]$ $F (x) = \frac{x^{2 n}}{{(x^{2} + 1)}^{n}} w i t h n f r o m N a n d n > 0 .$
	Answered by sma3l2996 last updated on 24/Feb/18

	$t^{2} = x^{2} + 1$ $F = \frac{{(t^{2} - 1)}^{n}}{t^{2 n}}$ $F = {(1 - \frac{1}{t^{2}})}^{n} = \underset{k = 0}{\sum^{n}} C_{n}^{k} {(- \frac{1}{t^{2}})}^{k}$ $F (x) = \underset{k = 0}{\sum^{n}} C_{n}^{k} {(- \frac{1}{x^{2} + 1})}^{k}$
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