Validity of binomial expansion for any power I know that
$$(1+x)^n=1+nx+\frac {n(n-1)}{2!}x^2+\frac {n(n-1)(n-2)}{3!}x^3+\frac {n(n-1)(n-2)(n-3)}{4!}x^2+\\
\dots+\frac {n(n-1)\dots(n-r+1)}{r!}x^r\text.$$
I know that this formula is valid for any real number n, provided that $|x|<1$. The formula extends infinitely for a negative value of $n$, and terminates for a positive value of $n$. Is there a formula for a case where $|x|>1$? Why is this formula applicable only for $|x|<1$?

The series you have written is valid for any real value of $x$ if $n$ is zero or a natural number. In this case your expansion is an identity and you get a polynomial of degree $n+1$ in $x$.
But if $n$ is negative integer or non integral, your expansion is valid only for |x|<1 and the series is infinite and convergent.
For example
$$(1+x)^4=1+4x+6x^2+4x^3+x^4.$$
$$(1+x)^{4.3}=1+4.3x+(4.3*3.3)x^2/2+(4.3*3.3*3.2)x^3/6+........$$
$$(1+x)^{-4.3}=1-4.3x+(4.3*5.3)x^2/2-(4.3*5.3*6.3)x^3/6+........$$
$$(1-x)^{-1}=1+x+x^2+x^3+......., \text{if}~ |x|<1$$
$$(1+1/3)^{1/2}=1+\frac{1}{2}(1/3)-\frac{1.1}{2.2.2!}(1/3)^2+\frac{1.1.3}{2.2.2.3!}(1/3)^3+.....$$
