# Partial Fraction Expansion

The expression

[s^2 + s - 1] / [s(s + 1)] + 1/s

Consider 1st term

[s^2 + s - 1] / [s(s + 1)^2]   =   A/s + B/(s + 1) + C/(s + 1)^2  ------------> (T)

Take the LCM

[s^2 + s - 1] / [s(s + 1)^2]   =   { A[s + 1)^2] + B[s(s + 1)] + Cs } / [ s(s + 1)^2 ]

[s^2 + s - 1]   =   { A[s^2 + 2s + 1)] + B(s^2 + s) + Cs }

[s^2 + s - 1]   =    As^2 + 2As + A + Bs^2 + Bs + Cs

[s^2 + s - 1]   =    (A + B)s^2 + (2A + B + C)s + A

Compare the coefficients of S^2, s and constant terms

A   =   - 1

A + B   =   1  -----------------------------> (1)

2A + B + C   =   1  --------------------------> (2)

Substitute A  = - 1 in equation (1)

-1 + B   =   1

B   =   1 + 1

B   =   2

Substitute A  = - 1 and B  =  2 in equation (2)

2(-1) + 2 + C   =   1

-2 + 2 + C   =   1

C   =   1

Substitute A  = - 1, B  =  2 and C  =  1 in equation (T)

[s^2 + s - 1] / [s(s + 1)^2]   =   (-1/s) + 2/(s + 1) + 1/(s + 1)^2

Hence

[s^2 + s - 1] / [s(s + 1)] + 1/s   =   [ (-1/s) + 2/(s + 1) + 1/(s + 1)^2 ] + 1/s

[s^2 + s - 1] / [s(s + 1)] + 1/s   =    2/(s + 1) + 1/(s + 1)^2