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[✓] Verify a symbolic equality?

GROUPS:

Hello everyone. Is it possible to verify the following identity in Mathematica?

Sum[Subscript[b, m] z^(-m), {m, 0, M}]/
  Sum[Subscript[a, n] z^(-n), {n, 0, N}] == (Subscript[b, 0]/
    Subscript[a, 
     0]) (Sum[(Subscript[b, m]/Subscript[b, 0]) z^(-m), {m, 0, M}]/
    Sum[(Subscript[a, n]/Subscript[a, 0]) z^(-n), {n, 0, N}])

With == the output is not True or False, it's the same as input.

Thank you for your help.

POSTED BY: Gennaro Arguzzi
Answer
3 months ago

Sum does not automatically collect common coefficients, but you can do it with a replacement rule:

collectCoefficientInSum := 
  Sum[(coeff_)*x_, {y_, z__}] /; FreeQ[coeff, y] :> 
   coeff*Sum[x, {y, z}];
(Sum[Subscript[b, m] z^(-m), {m, 0, M}]/
    Sum[Subscript[a, n] z^(-n), {n, 0, N}] == (Subscript[b, 0]/
      Subscript[a, 
       0]) (Sum[(Subscript[b, m]/Subscript[b, 0]) z^(-m), {m, 0, M}]/
      Sum[(Subscript[a, n]/Subscript[a, 0]) z^(-n), {n, 0, 
        N}])) /.
 collectCoefficientInSum
POSTED BY: Gianluca Gorni
Answer
3 months ago

Group Abstract Group Abstract