This proof starts out by considering the differential operator which is spreadable across addition . And:

, which is important for this proof.

And next, using traditional fractional calculus laws :

which comes to, (if you want me to show you the long work out just ask, I'm trying to be brief)

where is the digamma function.

So now we do the fun part:

So we just plug in our formula for and divide it by n!:

(divide by n!.)

we expand these and seperate and rearrange:

And now if you're confused what t represents, you'll be happy to hear we eliminate it now by setting it to equal 0. therefore, all our gammas are factorials and the left hand side becomes e^x by ln(x) and the since digamma function for integers arguments can be expressed through harmonic numbers where is the euler/mascheroni constant:

I've been unable to properly do the ratio test but using Pari gp it seems to converge for values x >e, but failed at 1000, worked for 900.

I decided to multiply the infinite series and I got:

Using Pari gp nothing seems to converge, but that may be fault to may coding.

I'm wondering, has anybody seen this before? And I am also mainly wondering how I can prove the radius of convergence of this. Also, I wonder if this could further suggest the gamma function as the natural extension of the factorial function, since these values do converge.

, which is important for this proof.

And next, using traditional fractional calculus laws :

which comes to, (if you want me to show you the long work out just ask, I'm trying to be brief)

where is the digamma function.

So now we do the fun part:

So we just plug in our formula for and divide it by n!:

(divide by n!.)

we expand these and seperate and rearrange:

And now if you're confused what t represents, you'll be happy to hear we eliminate it now by setting it to equal 0. therefore, all our gammas are factorials and the left hand side becomes e^x by ln(x) and the since digamma function for integers arguments can be expressed through harmonic numbers where is the euler/mascheroni constant:

I've been unable to properly do the ratio test but using Pari gp it seems to converge for values x >e, but failed at 1000, worked for 900.

I decided to multiply the infinite series and I got:

Using Pari gp nothing seems to converge, but that may be fault to may coding.

I'm wondering, has anybody seen this before? And I am also mainly wondering how I can prove the radius of convergence of this. Also, I wonder if this could further suggest the gamma function as the natural extension of the factorial function, since these values do converge.