
theorem Th26:
  for b be Nat st b > 1 holds
  powerfact b is summable & Sum powerfact b <= b/(b - 1)
  proof
    let b be Nat;
    assume
A1: b > 1; then
    |. 1/b .| < 1/1 by XREAL_1:76; then
A3: (1/b) GeoSeq is summable by SERIES_1:24;
A2: for i be Nat holds 0 <= (powerfact b).i
    proof
      let i be Nat;
      (powerfact b).i = 1/(b to_power (i!)) by DefPower;
      hence thesis;
    end;
A4: for i be Nat holds (powerfact b).i <= ((1/b) GeoSeq).i by A1,Th25;
    then
    Sum powerfact b <= Sum ((1/b) GeoSeq) by A2,A3,SERIES_1:20;
    hence thesis by Th13,A1,A2,A3,A4,SERIES_1:20;
  end;
