Understanding the Binomial Series Expansion and Its Integration

In this article, we explore a fascinating mathematical problem that involves the binomial series expansion and its relation to integral calculus. Specifically, we will delve into the process of solving the sum and integral representations of a function, and how they lead to the double factorial notation.

The Mathematical Problem: A Sum and Integral Connection

Consider the following mathematical expression:

[sum_{r0}^{n} frac{(-1)^{r} binom{n}{r}}{2r 1}]

This expression is intriguing as it connects the binomial coefficients with an alternating series and is related to an integral:

[int_{0}^{1} (1 - x^2)^n dx]

Understanding the Integral Representation

The integral expression (I_n int_{0}^{1} (1 - x^2)^n dx) is a classic problem in calculus. To solve this, we start by expanding the integrand using the binomial theorem:

[ (1 - x^2)^n 1 - nC1 x^2 nC2 x^4 - ... (-1)^n nCn x^{2n}]

Integrating this expression term by term from 0 to 1, we find:

[int_{0}^{1} (1 - x^2)^n dx 1 - frac{nC1}{2} frac{nC2}{4} - frac{nC3}{6} ... (-1)^n frac{nCn}{2n}]

Solving the Integral Using Integration by Parts

To further simplify the integral, we apply the integration by parts method. Let's denote:

[I_n int_{0}^{1} (1 - x^2)^n dx]

We can express (I_n) in terms of (I_{n-1}) as follows:

[I_n I_{n-1} - frac{1}{2n} I_n]

Solving for (I_n), we get:

[I_n frac{2n}{2n 1} I_{n-1}]

Using the base case (I_0 1), we can recursively find:

[I_n frac{2n cdot 2n-2 cdot ... cdot 4 cdot 2}{(2n 1) cdot 2n-1 cdot ... cdot 5 cdot 3} frac{2n!!}{(2n 1)!!}]

Connecting Sum and Integral

The expression (I_n) is the same as the given sum:

[sum_{r0}^{n} frac{(-1)^{r} binom{n}{r}}{2r 1} frac{2n!!}{(2n 1)!!}]

This elegant result is achieved through the clever use of integral calculus and the binomial theorem. It highlights the deep connection between discrete series and continuous integrals, and the power of the double factorial notation in compactly representing such expressions.

Keywords: Binomial Series, Integral Calculus, Double Factorial