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1/(1-x)^k

I didn't understand the circled part in the attached photo ?

Is it binomial ? However, -k may be negative and there is no negative number in binomial theory ?
question.png
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codeBuilder
Asked:
codeBuilder
1 Solution
 
aadihCommented:
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ChloesDadCommented:
yes it is binomial, k is a positive integer.

Wiki has some information about negative binomials

http://en.wikipedia.org/wiki/Binomial_coefficient
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GwynforWebCommented:
if Bin(k,n) = k!/(n!(k-n)!

then

Bin(-k,n) =(-1)^n *k!/(n!(k-n)!

               = (-1)^n *Bin(k,n)

So for instance  

  1/(1-x) = 1+ x + x² + x³ + ....

To really understand why this is the case requires going beyond Pascal's triangle. These ideas are often presented without sufficient justification, in essence they are saying:- believe me it is true and it works.

The coefficients for -ve and fractional powers come from looking at the problem as a Taylor Series expansion.
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codeBuilderAuthor Commented:
Bin(k,n) = k!/(n!(k-n)!
Bin(-k,n) =(-1)^n *k!/(n!(k-n)!


Each has one paranthesis which don't have any mathcing paranthesis?
Can you fix it please @GwynforWeb ?
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GwynforWebCommented:
Sorry it is this

Bin(k,n) = k!/(n!(k-n)!)
Bin(-k,n) =(-1)^n *k!/(n!(k-n)!)
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