For any real alpha and |x| < 1: (1+x)^alpha = $sum_{r=0}^{infinity}$ C(alpha,r)$x^r$, where the generalized binomial coefficient is C(alpha,r) = alpha(alpha-1)...$\frac{alpha-r+1}{r}$!.

When alpha is a positive integer, the series terminates at r = alpha (recovering the standard theorem). For non-integer or negative alpha, it is infinite.

Key expansions for JEE: (1-x)^{-1} = 1 + x + $x^2$ + $x^3$ + ... (geometric series) (1-x)^{-2} = 1 + 2x + 3$x^2$ + 4$x^3$ + ... (coefficients are r+1) (1-x)^{-n} = sum C(n+r-1, r)*$x^r$ (negative binomial) (1+x)^{1/2} = 1 + x/2 - $x^2$/8 + $x^3$/16 - ... (square root expansion)

The negative binomial series (1-x)^{-n} = sum C(n+r-1,r)*$x^r$ is the most used in JEE. The coefficient C(n+r-1,r) can also be written as C(n+r-1, n-1).

Approximation: For small |x|, (1+x)^alpha approximately equals 1 + alphax. Second order: 1 + alphax + alpha(alpha-1)*$x^2$/2. These are used to approximate values like sqrt(1.02) or (0.98)^{1/3}.

Convergence: The series converges for |x| < 1 regardless of alpha. At x = +/-1, convergence depends on alpha (converges at x=1 if alpha > -1; converges at x=-1 if alpha > 0). JEE problems rarely test boundary convergence.