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1/(1-x)^k

Posted on 2013-05-24
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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 ?
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Question by:codeBuilder
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LVL 24

Expert Comment

ID: 39195760
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LVL 15

Expert Comment

ID: 39195777
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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LVL 31

Expert Comment

ID: 39195827
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 + xÂ˛ + xÂł + ....

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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Author Comment

ID: 39195884
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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LVL 31

Accepted Solution

GwynforWeb earned 2000 total points
ID: 39196035
Sorry it is this

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