Binomial Expansion of 1/sqrt(1-x^2)
Introduction
The binomial theorem is a fundamental concept in algebra, which provides a formula for expanding powers of a binomial expression. In this article, we will discuss the binomial expansion of 1/sqrt(1-x^2), a special case of the binomial theorem.
Binomial Theorem
The binomial theorem states that for any positive integer n,
$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k$
where ${n \choose k}$ is the binomial coefficient.
Expansion of 1/sqrt(1-x^2)
To expand 1/sqrt(1-x^2), we can use the binomial theorem with a = 1 and b = -x^2. Then,
$\frac{1}{\sqrt{1-x^2}} = (1-x^2)^{-1/2} = \sum_{k=0}^{\infty} {-1/2 \choose k} (-x^2)^k$
Using the formula for binomial coefficients, we get
${-1/2 \choose k} = \frac{(-1/2)(-1/2-1)(-1/2-2)\cdots(-1/2-k+1)}{k!} = \frac{(-1)^k \Gamma(k+1/2)}{k! \Gamma(1/2)}$
where Γ(z) is the gamma function.
Substituting this back into the expansion, we get
$\frac{1}{\sqrt{1-x^2}} = \sum_{k=0}^{\infty} \frac{(-1)^k \Gamma(k+1/2)}{k! \Gamma(1/2)} x^{2k}$
Simplification
Using the properties of the gamma function, we can simplify the coefficients:
$\frac{(-1)^k \Gamma(k+1/2)}{k! \Gamma(1/2)} = \frac{(-1)^k (2k-1)!!}{(2k)!!}$
where !! denotes the double factorial.
Thus, the final expansion is
$\frac{1}{\sqrt{1-x^2}} = \sum_{k=0}^{\infty} \frac{(-1)^k (2k-1)!!}{(2k)!!} x^{2k}$
This expansion is valid for |x| < 1.
Conclusion
In this article, we have derived the binomial expansion of 1/sqrt(1-x^2) using the binomial theorem. The resulting expansion is a power series in x, with coefficients involving the gamma function and double factorials. This expansion has applications in various fields, such as calculus, algebra, and analysis.
