Partial Fraction Decomposition

algebra analysis

Definition

Partial Fraction Decomposition

The partial fraction decomposition of a rational fraction is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or serveral fractions with a simpler denominator.

$$\frac{f(x)}{g(x)}=p(x)+\sum _j\frac{f_j(x)}{g_j(x)}$$

where $p(x)$ is a polynomial, and, for each $j$, the denominator $g_j(x)$ is a power of an irreducible polynomial.

Examples

Example: $\frac{1}{n(n+1)}$

$$\begin{array}{rl}\forall n:\frac{1}{n(n+1)}& =\frac{A}{n}+\frac{B}{n+1}\\ 1& =\frac{An(n+1)}{n}+\frac{Bn(n+1)}{n+1}\\ 1& =A(n+1)+Bn\\ 1& =An+A+Bn\\ 1& =n(A+B)+A\end{array}$$

If $n=0$, then $A=1$ and $B=-1$ to fulfil this equation for all $n$. Therefore:

$$\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$

which is a more helpful representation for telescoping series.