MHB Binomial Expansion - Fractional Powers

[Mathick]

Hello!

We know from 'Binomial Expansion' that [math](1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+...[/math] for [math] \left| x \right|<1 [/math]. Why doesn't it work for other values of [math]x[/math]? I can't understand this condition. I would be really grateful for clear explanation!

 

[kaliprasad]

Hello!

We know from 'Binomial Expansion' that [math](1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+...[/math] for [math] \left| x \right|<1 [/math]. Why doesn't it work for other values of [math]x[/math]? I can't understand this condition. I would be really grateful for clear explanation!

when n is integer after n+1 terms the numerator becomes zero and so Binomial Expansion holds. for any x
but when n is not an integer there are infinite terms and if |x| is 1 or more then the series diverges and for |x| < 1 this converges as x^n tends to be zero.