Can You Find a Finite Expression for the Summation of Factorial Terms?

  • Context: Graduate 
  • Thread starter Thread starter Atran
  • Start date Start date
  • Tags Tags
    Factorial Summation
Click For Summary

Discussion Overview

The discussion revolves around finding a finite expression for the summation of factorial terms, specifically the series defined as the sum of factorials from 0 to m. Participants explore various expressions and approaches to represent this summation in a more compact form.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant presents an expression for the summation of factorials, suggesting a nested function form for f(m).
  • Another participant shares a link to the Online Encyclopedia of Integer Sequences (OEIS), possibly indicating a connection to known sequences related to the factorial summation.
  • A third participant extends the function f(x) to non-integer arguments using an integral representation, providing numerical values for specific non-integer inputs.
  • A fourth participant proposes several functions (f0, f1, f2) that compute the sum of factorials using different programming approaches.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on a single finite expression for the summation of factorial terms, and multiple competing approaches are presented without resolution.

Contextual Notes

The discussion includes various mathematical representations and numerical approximations, but the assumptions underlying the proposed functions and the accuracy of numerical integration are not fully explored.

Who May Find This Useful

Readers interested in mathematical series, factorial functions, or numerical methods may find the discussion relevant.

Atran
Messages
93
Reaction score
1
Hi, I've been trying to make another expression for,

m
[tex]\sum[/tex](k!) = f(m)
k=0

I did one expression that is,

f(m) = 2(3(4(5...((m-1)(m+1)+1)...+1)+1)+1)+1

For instance, f(4) = 2(3(4+1)+1)+1

Can you hint me on finding a finite expression for the above expression?
Thanks for help.
 
Mathematics news on Phys.org
FYI your function f(x) can be extended to non-integer argument x as follows:

Since

s! = integral(0,infinity)[ x^s e^-x dx]

it follows that

f(s) = integral(0,infinity)[ (1+x+x^2+...+x^s) e^-x dx]
= integral(0,infinity)[ (1-x^(s+1))/(1-x) *e^-x dx]

Numerical integration (not sure how accurate) yields
f(-1.5)=-1.21027
f(-0.5) = 0.562187
f(0.5) = 1.44841
f(1.5) = 2.77775
f(2.5) = 6.1011
f(i) = 0.963332 + 0.751815i
 
Last edited:
f0(n)= sum(i=0, n-1, i!)
f1(n)= my(r=1,s=1); for(i=1,n-1,s*=i;r+=s);r
f2(n)= my(r=n);forstep(i=n-2,1,-1,r=r*i+1);r
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 10 ·
Replies
10
Views
3K
  • · Replies 8 ·
Replies
8
Views
3K