A formula of the product of the first n integers?

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Discussion Overview

The discussion revolves around the existence of a formula for the product of the first n integers, akin to the formula for the sum of the first n integers. Participants explore whether such a formula exists and discuss related concepts, including factorials and approximations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses a desire for a formula for the product of the first n integers, suggesting it would be helpful.
  • Another participant mentions the factorial notation n! as a representation of the product of the first n integers.
  • A participant questions whether there is a formula similar to n(n+1)/2 for sums, proposing that there might not be one since n! is commonly used.
  • Stirling's approximation to n! is introduced as a useful approximation for large n.
  • Another participant prefers Gosper's reformulation of Stirling's formula for approximation purposes.
  • A participant brings up the integral form of the factorial function, noting its potential relevance but expressing doubt about its applicability to the original question.
  • There is a clarification regarding the formula for the sum of integers, with a participant noting the distinction between starting at 1 versus 0.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a formula for the product of the first n integers. Multiple viewpoints are presented, including the use of factorials and approximations, indicating ongoing debate.

Contextual Notes

Some assumptions about the definitions of the terms and the context of the discussion may be missing, particularly regarding the applicability of Stirling's approximation and the integral form of the factorial.

quasar987
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I'm sure it exists, and it'd help me to have it. Thx!
 
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n!

......
 
No no, like there's n(n+1)/2 for the sum of the first n integers. Is there an equivalent formula for the product?
 
quasar987 said:
No no, like there's n(n+1)/2 for the sum of the first n integers. Is there an equivalent formula for the product?

I want to say no since if there was we would probably use that instead of n! everywhere, but there's stirling's approximation to n!.
 
Last edited:
You're probably right.
 
Stirling's formula gives a good approximation. I prefer to use Gosper's reformulation of same.
 
Well, there's always the integral form of the factorial function; although I doubt it's what you're looking for:

n! = \Gamma(n+1) = \int_0^{\infty} e^{-t} t^n dt
 
it's n(n+1)/2 if you start at 1 and n(n-1)/2 if you start at zero if I recall...
 
Robokapp said:
it's n(n+1)/2 if you start at 1 and n(n-1)/2 if you start at zero if I recall...

I think you mean n and n-1, respectively, if you're talking about sums. Obviously adding in 0 has no effect.
 

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