MHB Binomial Expansion - Fractional Powers

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The Binomial Expansion formula (1+x)^n is valid for |x|<1 due to convergence of the series. When n is an integer, the series terminates after n+1 terms, making it applicable for any x. However, for non-integer n, the series has infinite terms, leading to divergence when |x| is 1 or greater. The condition |x|<1 ensures that x^n approaches zero, allowing the series to converge. Therefore, the restriction on x is crucial for the validity of the Binomial Expansion with fractional powers.
Mathick
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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!
 
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Mathick said:
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.
 
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