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 Th21:
  n>0 implies cseq(n) is summable & Sum(cseq(n))=(1+(1/n)) ^ n &
  Sum(cseq(n))=dseq.n
proof
A1: now
    let k;
    assume k>=n+1;
    then
A2: k>n by NAT_1:13;
    thus cseq(n).k = (n choose k)*(n ^ (-k)) by Def3
      .= 0*(n ^ (-k)) by A2,NEWTON:def 3
      .= 0;
  end;
  assume
A3: n>0;
A4: now
    let k;
    assume k<n+1;
    then k<=n by NAT_1:13;
    hence cseq(n).k=((1,1/n) In_Power n).(k+1) by A3,Th20;
  end;
A5: len((1,1/n) In_Power n)=n+1 by NEWTON:def 4;
  hence cseq(n) is summable by A4,A1,Th18;
  thus (1+(1/n)) ^ n = Sum((1,1/n) In_Power n) by NEWTON:30
    .= Sum(cseq(n)) by A5,A4,A1,Th18;
  hence thesis by Def4;
end;
