Binomial Expansion of (1+x)^3/2
The binomial theorem allows us to expand expressions of the form (a + b)^n, where n is a rational number. In this case, we are interested in the expansion of (1+x)^3/2.
The Binomial Theorem
The binomial theorem states:
(a + b)^n = a^n + \binom{n}{1}a^(n-1)b + \binom{n}{2}a^(n-2)b^2 + ... + \binom{n}{r}a^(n-r)b^r + ... + b^n
where:
- n is a rational number
- r is an integer between 0 and n
- \binom{n}{r} is the binomial coefficient, calculated as n! / (r! * (n-r)!)
Expanding (1+x)^3/2
Let's apply the binomial theorem to our expression:
(1 + x)^3/2 = 1^(3/2) + \binom{3/2}{1}1^(1/2)x + \binom{3/2}{2}1^(-1/2)x^2 + \binom{3/2}{3}1^(-3/2)x^3 + ...
Now, let's calculate the binomial coefficients:
- \binom{3/2}{1} = (3/2)! / (1! * (1/2)!) = 3/2
- \binom{3/2}{2} = (3/2)! / (2! * (1/2)!) = 3/8
- \binom{3/2}{3} = (3/2)! / (3! * (-1/2)!) = 1/16
Substituting the coefficients back into the expansion:
(1 + x)^3/2 = 1 + (3/2)x + (3/8)x^2 + (1/16)x^3 + ...
Important Note:
The expansion of (1+x)^3/2 is an infinite series. This means that the terms continue indefinitely. However, the series converges for |x| < 1. This means that the expansion is valid for values of x between -1 and 1.
Applications
The binomial expansion of (1+x)^3/2 has applications in various fields including:
- Calculus: Used for approximating functions and finding their derivatives.
- Probability: Useful in calculating probabilities related to binomial distributions.
- Physics: Used in modeling physical phenomena like wave propagation.
By understanding the binomial expansion, we gain a powerful tool for working with expressions involving fractional exponents and can unlock a world of mathematical possibilities.
