Is the Binomial Theorem Applicable to All Real Numbers?

[prasannapakkiam]

I learned the Binomial Theorem a while ago. But it is only now that I think about how it is only useful for powers that are natural numbers. Can it be extended to all real numbers - e.g. 1/2?

 

[Werg22]

Yes, but in the form of infinite series.

 

[prasannapakkiam]

Can you please expand on this?

 

[Crosson]

Do you know about the Gamma function that extends the domain of the factorial from the integers to the positive real line?

 

[Werg22]

The expansion of a+b raised to a fractional power takes the form of the sum of an infinite number of terms, or more like the limit of a sum as you add more terms. What else could be expected anyway? An expansion of any sort has to be an infinite series to take into account irrational numbers. You can read a bit about it here:

http://www.mathsrevision.net/alevel/pages.php?page=4"

 

[prasannapakkiam]

thanks for the help.

 

[Gib Z]

This might also help :

http://tutorial.math.lamar.edu/AllBrowsers/2414/BinomialSeries.asp

Werg22's link has pretty much the same thing though, both helpful anyway :)