If I type


I would like Mathematica to return something like


or maybe


But it doesn't seem to want to do this (maybe because it's worried that the difference makes no sense if the sums don't converge?). Is there a simple way to let Mathematica know I would like it to make this kind of simplification?

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    $\begingroup$ A dirty way :ClearAll[s]; s /: s[f[x], {x, a_, b_}] - s[f[x], {x, c_, b_}] := s[f[x], {x, c + 1, a}]; simp[x_] := (x /. Sum -> s /. s -> Sum); simp[Sum[f[x], {x, m + 3, Infinity}] - Sum[f[x], {x, m, Infinity}]] $\endgroup$ – Dr. belisarius Mar 2 '14 at 22:58
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    $\begingroup$ Don't you want -Sum[f[x], {x, m, m+2}] instead? $\endgroup$ – Rahul Mar 3 '14 at 1:32
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    $\begingroup$ @RahulNarain: Indeed. I changed the example midway through posting and failed to change the second line. Going back to fix this now. $\endgroup$ – WillO Mar 3 '14 at 4:38
  • $\begingroup$ I searched the DifferenceDelta and DiscreteShift, but I do not think it has anything to do with it. $\endgroup$ – LCarvalho Jul 17 '17 at 13:58

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