Get edited series expansions with the Wolfram Language?

Questions
1

It is very easy using Series to get series expansion but the output is never in the way we see them in math textbooks where they are more understandable.
Il’ take take 2 examples

e^{x}~=~1+~x~+\frac{x^{2}}{2!}~+~\frac{x^{3}}{3!}~+~~\frac{x^{3}}{4!}~+....

\arcsin x~=~x~+~\frac{x^{3}}{2 \cdot3}~+~\frac{1 \cdot 3 }{2 \cdot4 \cdot 5}~ x^{5}~+~\frac{1 \cdot 3 \cdot5}{2 \cdot4 \cdot 6 \cdot7}~ x^{7}~+~....

I tried to get the above output using MMA, with not much success

My first idea was to start with the summation of the general term of a MacLaurin series

gterm[f_, x_, n_] :=  x^(n - 1) (D[f[x], {x, n - 1}] /. x -> 0)/(n - 1)!
exp = Sum[gterm[Exp, x, k], {k, 1, 5}]

No way to put a Hold or HoldForm somewhere in the definition of my function gterm to prevent the evaluation of the factorial.

So I started toying then with patterns and I came out with something which works with e^x but leaves me unsastified as hardly generalisable.

fx[a_] := 
Module[{}, maj = Cases[a, Times[_, Power[x, e_]] :> HoldForm[e!]]; 
pos = Position[a, _Rational]; ReplacePart[a, Thread[pos -> 1/maj]]]
exp //  fx

And my knowdledge of MMA is not good enough to handle the ArcSin case in such a way ..

Is there a better solution than the one I’m not satisfied with and could be adapted with minimal effort to handle more elaborate cases ?
Thanks.

2

A couple of attempts using Inactive:

Inactive[Plus] @@ 
 Table[Inactive[Times][SeriesCoefficient[Exp[x], {x, 0, k}], 
   Power[x, k]], {k, 0, 5}]
Inactive[Plus] @@ 
 Table[Times[SeriesCoefficient[Exp[x], {x, 0, k}], Power[x, k]], {k, 
   0, 5}]

The formatting is less satisfying when the coefficients are negative.

3

Hello Gianluca

Your answers evaluate the factorials and I don’t want this to take place. For Exp[x] I get 1+x+x^2/2+x^3/6+x^4/24+x^5/120 .

You gave me anyway an idea by using this rather new MMA function that I didn’t know about until I read you.

genterm = SeriesCoefficient[Exp[x] , {x, 0, n}][[1, 1, 1]] x^  n /. {Factorial[n] -> Inactivate[Factorial[n]]} (* general term for Exp[x]  MacLaurin expansion*)
Sum[genterm , {n, 0, 5}]

\frac{1}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}
Easy to finish it by simply rep[lacing the two first terms by 1 + x

However this is not generalisable to ArcSin[x] because I want here the factorial in the denominator of the general term to be simplified in the way you see it in most textbooks and my course materials too.

  (x^n Pochhammer[1/2, 1/2 (-1 + n)])/(n (1/2 (-1 + n))!) /. {n -> 5}

gives me \frac{3 x^{5}}{40} and I would like to have it expanded as \frac{1 \cdot 3}{2 \cdot 4 \cdot 5}~ x^{5}