text: # Partial Fraction Expansion
  
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  asked Dec 26, 2017
  
  ## 1 Answer
  
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  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
  
  answered Dec 27, 2017