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How do you compute $\dfrac{x}{(1+x)^2}$ using partial fractions? The reason I ask is because when I try to solve it I keep getting an impossible $A, B$.

$A(1+x) + B(1+x) = x$

$A + Ax + B + Bx = x$

$(A+B)x = 1$

$(A+B) = 0$

However a practice problem I'm working on implies this can be solved by partial fractions, but doesn't go into detail.

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  • $\begingroup$ This might be helpful. $\endgroup$ – Git Gud May 10 '13 at 6:25
  • $\begingroup$ The accepted answer here, has a generalized argument. $\endgroup$ – Inceptio May 19 '13 at 15:15
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In general, whenever you have an expression of the form $$\dfrac{P(x)}{(x+a)^n}$$ where $P(x)$ is of degree less than $n$ and you want to write it into partial fraction, then you need to proceed as follows. $$\dfrac{P(x)}{(x + a)^n} = \dfrac{A_1}{x+a} + \dfrac{A_2}{(x+a)^2} + \dfrac{A_3}{(x+a)^3} + \cdots + \dfrac{A_{n-1}}{(x+a)^{n-1}} + \dfrac{A_n}{(x+a)^n} \tag{$\star$}$$ and then find the coefficients $A_1, A_2,\ldots, A_n$. Note that from $(\star)$, we have $$P(x) = A_1(x+a)^{n-1} + A_2 (x+a)^{n-2} + A_3(x+a)^{n-3} + \cdots + A_n$$ From this, we have $A_n = P(-a)$ and in general, $$A_k = \left.\dfrac{d^{n-k} P(x)}{dx} \right \vert_{x=-a}$$

In your case, though it is relatively simple since $$\dfrac{x}{(1+x)^2} = \dfrac{1+x-1}{(1+x)^2} = \dfrac1{(1+x)} - \dfrac1{(1+x)^2}$$

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    $\begingroup$ Generalization always makes a good taste. +1 $\endgroup$ – mrs May 10 '13 at 6:29
  • $\begingroup$ @Git Gud: You can always edit the answer of some user, so just give it a shot. Btw, that error just a small LaTeX typo, no big deal. $\endgroup$ – user49685 May 10 '13 at 7:01
  • $\begingroup$ @user17762: Great answer, (+1). Do you have a name? Just curious $\endgroup$ – Inceptio May 10 '13 at 7:42
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Raskolnikov's answer is a pretty slick way to do this, but if you wanted a more general algorithm for problems like this, here you go.

Whenever the denominator is some linear term squared, like it is here, since it's $(1 + x)^2$, then you set up the partial fraction this way:

$$\frac{x}{(1 + x)^2} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2}$$

Then solving this is just a matter of multiplying both sides by $(1 + x)^2$.

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

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$$\frac{x}{(1 + x)^2} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2}$$ $$x=A(1+x)+B $$ put $x=-1$ gives $B=-1$ and compare coefficient of x from both side gives $A=1$ so answer is $$\frac{x}{(1 + x)^2} = \frac{1}{1 + x} - \frac{1}{(1 + x)^2}$$

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