Is the Summation of the Binomial Theorem Equal to Zero?

[EngWiPy]

Hello,

All we know the Binomial theorem which may be stated mathematically as:

$$\left(x+y\right)^n=\sum_{k=0}^n{n\choose k}y^k\,x^{n-k}$$

Now suppose that we have the following mathematical expression:

$$\sum_{k=0}^{n}{n\choose k}\,(-1)^k$$

if we substitute x=1 and y=-1 in the first equation we get the second. Is that mean the second equation is essentially zero, since $$(1-1)^n=0$$??

Regards

 

[Moo Of Doom]

Yes, indeed, unless n = 0.

 

[EngWiPy]

Yes, indeed, unless n = 0.

Why? In the case that n = 0, what will be the answer? 1?

 

[slider142]

0⁰ is not well-defined and neither is 0Ck for any k <> 0 (although there are generalizations that extend the domain beyond the definition using just factorials).