Binomial Expansion - Fractional Powers

[Mathick]

Hello!

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

 

[kaliprasad]

Hello!

We know from 'Binomial Expansion' that \(\displaystyle (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+...\) for \(\displaystyle \left| x \right|<1 \). Why doesn't it work for other values of \(\displaystyle x\)? 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.