The following code takes an arbitrary number of terms, shows their compressed simplified form,
the summand (with k substituted for n), and the approximate numerical error relative to e.
Here TC represents the number of terms (from Newton's series) to be combined into a single term
of the new series while t sets the number of these new terms to be evaluated.

Compressing Series For (e^( )):

Clear[k]; TC = 1; t = 20;
numerat[1] = 1;
numerat[TC_] := 
        numerat[TC - 1]*(n + TC - 1) + 1;
denominat = (n + TC - 1)!;
summand = 
      Together[Expand[numerat[TC]/denominat]]
expressn = summand /. n -> (TC*k)
k = t; 
    acc = Floor[N[-Log[10, expressn]]] + 15;
Print["Error ≈ ", 
      N[E - N[Sum[expressn, {k, 0, t - 1}], acc], acc]]

Compressing Series For (e^x):

Clear[k, x]; TC = 1; t = 20;
numerat[1] = x^n;
numerat[CT_] := 
        numerat[TC - 1]*(n + TC - 1) + 
          x^(n + TC - 1);
denominat = (n + TC - 1)!;
summand = 
      Together[Expand[numerat[TC]/denominat]]
pwexpressn = summand /. n -> (TC*k)
x = 1/2;
k = t; 
    acc = Floor[N[-Log[10, pwexpressn]]] + 15;
Print["Error ≈ ", 
      N[E - N[Sum[pwexpressn, {k, 0, t - 1}]^(1/x), acc], 
        acc]]