Welcome to the advanced topic of Binomial Expansion for Rational Exponents! This is a crucial concept in JEE Mathematics, particularly for calculus and advanced algebra problems. While the standard binomial theorem applies to positive integer exponents, we'll explore how to extend this powerful tool to cases involving fractions and roots.

Recall the binomial theorem for a positive integer $n$:

$(x+a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. This formula provides a finite sum of terms.

When the exponent $n$ is not a positive integer (i.e., it's a rational number like $1/2$, $-3/4$, etc.), the factorial function $n!$ is not directly defined in the standard way. Furthermore, the sum becomes infinite. This requires a different approach, often referred to as the generalized binomial theorem or the binomial series.

For any real number $n$ and $|x| < 1$, the binomial expansion is given by the infinite series:

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + \frac{n(n-1)\dots(n-k+1)}{k!}x^k + \dots$

This can be written more compactly using the generalized binomial coefficient:

$(1+x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k$

where $\binom{n}{k} = \frac{n(n-1)\dots(n-k+1)}{k!}$.

Let's break down the generalized binomial coefficient $\binom{n}{k}$ for a rational exponent $n$:

$\binom{n}{k} = \frac{n(n-1)(n-2)\dots(n-k+1)}{k!}$

For example, if $n = 1/2$ and $k=2$, then:

$\binom{1/2}{2} = \frac{(1/2)(1/2 - 1)}{2!} = \frac{(1/2)(-1/2)}{2} = \frac{-1/4}{2} = -1/8$

And if $n = 1/2$ and $k=3$, then:

$\binom{1/2}{3} = \frac{(1/2)(1/2 - 1)(1/2 - 2)}{3!} = \frac{(1/2)(-1/2)(-3/2)}{6} = \frac{3/8}{6} = 1/16$

So, the expansion of $(1+x)^{1/2}$ starts as $1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 + \dots$

A critical aspect of the generalized binomial theorem is its convergence. The series converges if $|x| < 1$. If $n$ is a non-negative integer, the series terminates and is valid for all $x$. However, for rational or real exponents, the series is infinite and requires this condition for convergence. This is vital for applying the expansion in calculus, such as integration or differentiation.

This expansion is frequently used to approximate values of roots (like $\sqrt{1.02}$ or $\sqrt[3]{26}$), simplify expressions involving fractional powers, and solve problems in calculus involving integration of functions with fractional exponents. For instance, integrating $\frac{1}{\sqrt{1+x^2}}$ often involves using a binomial expansion.

To approximate $\sqrt{1.02}$, we can use the expansion of $(1+x)^n$ with $n = 1/2$ and $x = 0.02$. Since $|0.02| < 1$, the series converges.

$(1+0.02)^{1/2} = 1 + \frac{1}{2}(0.02) + \frac{(1/2)(1/2-1)}{2!}(0.02)^2 + \dots$

$= 1 + 0.01 + \frac{(1/2)(-1/2)}{2}(0.0004) + \dots$

$= 1 + 0.01 - \frac{1}{8}(0.0004) + \dots$

$= 1 + 0.01 - 0.00005 + \dots$

$= 1.00995$

This gives a very close approximation to the actual value of $\sqrt{1.02}$.

Mastering the binomial expansion for rational exponents involves understanding the generalized binomial coefficient, the infinite series representation, and the crucial convergence condition $|x|<1$. Practice applying it to approximation problems and calculus integration to solidify your understanding for JEE.