Explore Some Input Complete Classes of Sums

Sum can compute all sums for many classes of summands including polynomial, polynomial exponential, rational and rational exponential.

In[1]:=

Manipulate[

 With[{g = f /. n -> m}, 

  Text@TraditionalForm@Pane[g == ReleaseHold[g], 400]], {{m, 25, "n"},

   1, 50, 1}, {{f, HoldForm[Sum[k^n 3^k, k]], 

   "sum"}, {HoldForm[Sum[k^n, k]], HoldForm[Sum[k^n 3^k, k]], 

   HoldForm[Sum[1/(k (k + n)), k]], 

   HoldForm[Sum[2^k/(k (k + n)), k]]}}]

Out[1]=