
theorem Th23:
  for f be Real_Sequence, n be Nat st f is summable & f.0 = 0 holds
  Sum f = Sum FinSeq (f,n) + Sum (f ^\ (n+1))
  proof
    let f be Real_Sequence, n be Nat;
    assume
A1: f is summable & f.0 = 0; then
    Sum f = Partial_Sums(f).n + Sum (f ^\ (n+1)) by SERIES_1:15;
    hence thesis by A1,Th21;
  end;
