Binomial Series Simplification

  • Thread starter Thread starter courtrigrad
  • Start date Start date
  • Tags Tags
    Binomial Series
Click For Summary
SUMMARY

The discussion focuses on the simplification of the binomial coefficient \(\binom{-3}{n}\) using the formula \(\frac{(-3)(-4)(-5)\cdots[-(n+2)]}{n!}\). The key steps involve recognizing the pattern in the numerator, which leads to the expression \(\frac{(-1)^{n}(n+1)(n+2)}{2}\). The simplification process effectively demonstrates how to manipulate negative binomial coefficients into a more manageable form.

PREREQUISITES
  • Understanding of binomial coefficients and their properties
  • Familiarity with factorial notation and operations
  • Basic knowledge of algebraic manipulation
  • Concept of negative integers in combinatorial contexts
NEXT STEPS
  • Study the properties of negative binomial coefficients
  • Explore combinatorial identities involving binomial coefficients
  • Learn about generating functions and their applications in combinatorics
  • Investigate the relationship between binomial coefficients and polynomial expansions
USEFUL FOR

Mathematicians, students studying combinatorics, and anyone interested in advanced algebraic techniques related to binomial coefficients.

courtrigrad
Messages
1,236
Reaction score
2
How does one deduce the following:

We are given

[tex]\binom{-3}{n} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot\cdot(-3-n+1)}{n!} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot[-(n+2)]}{n!}[/tex]. How do we get from here to:
[tex]\frac{(-1)^{n}\cdot2\cdot3\cdot4\cdot5\cdot\cdot\cdot\cdot(n+1)(n+2)}{2\cdot n!} = \frac{(-1)^{n}(n+1)(n+2)}{2}[/tex]?Thanks
 
Last edited:
Physics news on Phys.org
nevermind. I figured it out
 

Similar threads

Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
  • · Replies 1 ·
Replies
1
Views
1K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##a\cdot b = b \cdot a ##using Peano postulates
  • · Replies 3 ·
Replies
3
Views
1K
Does each norm on vector space become discontinuous when restricted to S^1?
  • · Replies 1 ·
Replies
1
Views
1K
Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?
  • · Replies 6 ·
Replies
6
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
  • · Replies 2 ·
Replies
2
Views
1K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Show that the sequence converges
  • · Replies 9 ·
Replies
9
Views
2K
Checking series for convergence
  • · Replies 5 ·
Replies
5
Views
1K