# Manipulating infinite series

If I type

Sum[f[x],{x,m,Infinity}]-Sum[f[x],{x,m+3,Infinity}]


I would like Mathematica to return something like

Sum[f[x],{x,m,m+2}]


or maybe

f[m]+f[m+1]+f[m+2]


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?

• 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}]] – Dr. belisarius Mar 2 '14 at 22:58
• Don't you want -Sum[f[x], {x, m, m+2}] instead? – Rahul Mar 3 '14 at 1:32
• @RahulNarain: Indeed. I changed the example midway through posting and failed to change the second line. Going back to fix this now. – WillO Mar 3 '14 at 4:38
• I searched the DifferenceDelta and DiscreteShift, but I do not think it has anything to do with it. – LCarvalho Jul 17 '17 at 13:58