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Proving a formula with binomial coefficient when n=-1

  • Thread starter Seydlitz
  • Start date
263
4
1. Homework Statement

Prove that ##\binom{-1}{k}=(-1)^k##

3. The Attempt at a Solution

Using induction on ##k##,

##\binom{-1}{0}=1## which is true also for ##(-1)^0=1##

Assuming ##\binom{-1}{k}=(-1)^k##, then ##\binom{-1}{k+1}=(-1)^{k+1}##

Indeed when ##n=-1##, we can write rewrite this ##\frac{n!}{k!(n-k)!}## as ##\frac{(-1)^k(k)!}{k!}## to avoid negative factorial. Hence ##\binom{-1}{k+1}=\frac{(-1)^k(k+1)!}{(k+1)!}=(-1)^{k+1}##.

##\blacksquare##

I just want to confirm if my proof by induction method is valid.

Thank You
 
263
4
You should recast the combinatorials in terms of gamma functions before your start.

See http://mathworld.wolfram.com/BinomialCoefficient.html
This exercise is taken from Boas Mathematical Methods in Physics Chapter 1, the readers haven't been exposed to Gamma function nor does the problem actually requires one to prove the statement. It only requires one to show but I just want to prove it if possible using simple induction.
 

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