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)...(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 + 3x^2 + 4x^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.