reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th39:
  n>0 & x=1/(n+1) implies n!*Sum(eseq^\(n+1))<=x/(1-x)
proof
  assume that
A1: n>0 and
A2: x=1/(n+1);
  deffunc F(Nat)=x ^ ($1+1);
  consider seq being Real_Sequence such that
A3: for k holds seq.k=F(k) from SEQ_1:sch 1;
A4: now
    let k;
A5: (n!(#)(eseq^\(n+1))).k = n!*((eseq^\(n+1)).k) by SEQ_1:9
      .= n!*eseq.(n+1+k) by NAT_1:def 3
      .= n!*(1/((n+k+1)!)) by Def5
      .= n!/((n+k+1)!);
    hence (n!(#)(eseq^\(n+1))).k>=0;
    seq.k=x ^ (k+1) by A3;
    hence (n!(#)(eseq^\(n+1))).k<=seq.k by A2,A5,Th38;
  end;
A6: seq.0 = x ^ (0+1) by A3
    .= x;
A7: eseq^\(n+1) is summable by Th23,SERIES_1:12;
  n+1>0+1 by A1,XREAL_1:6;
  then
A8: x<1 by A2,XREAL_1:212;
A9: now
    let k;
    thus seq.(k+1) = x ^ (k+1+1) by A3
      .= x ^ 1*(x^(k+1)) by A2,POWER:27
      .= x*(x^(k+1))
      .= x*seq.k by A3;
  end;
  |.x.|=x by A2,ABSVALUE:def 1;
  then seq is summable by A8,A9,SERIES_1:25;
  then
A10: Sum(n!(#)(eseq^\(n+1)))<=Sum(seq) by A4,SERIES_1:20;
  |.x.|<1 by A2,A8,ABSVALUE:def 1;
  then Sum(seq)=x/(1-x) by A6,A9,SERIES_1:25;
  hence thesis by A7,A10,SERIES_1:10;
end;
