Binomial series as an example of Maclaurin's series

An example of a Maclaurin series is the Binomial theorem for negative / fractional indices:

\[(1 + x)^{n} = 1 + nx + \frac{n(n-1)}{2!} x^{2} + \frac{n(n-1)(n-2)}{3!} x^{3} + \cdots \text{ for}\hspace{0.5em} |x| < 1\]

To see this, let us work through all the steps of calculating the Maclaurin's series
\[ f(x) = \sum_{n=0}^\infty c_n x^n \]
for the function
\[ f(x) = (1+x)^n, \]
where \(n \in \mathbb{R} - \mathbb{N}\).

We start with
\[ f(0) = (1+0)^n = 1. \]
So,
\[ c_0 = f(0) = 1. \]

Differentiating \(f\) with respect to \(x\):
\[ f^{(1)}(x) = n (1+x)^{n-1}. \]
So,
\[ f^{(1)}(0) = n(1+0)^{n-1} = n. \]
Thus,
\[ c_1 = f^{(1)}(0) = n. \]

Proceeding with further differentiation:
\[ f^{(2)}(x) = n(n-1)(1+x)^{n-2}. \]
Thus,
\[ f^{(2)}(0) = n(n-1). \]
So,
\[ c_2 = \frac{f^{(2)}(0)}{2!} = \frac{n(n-1)}{2}. \]

By induction, one has
\[ f^{(k)}(0) = n(n-1)(n-2) \cdots (n-k+1). \]
Thus,
\[ c_k = \frac{n(n-1)(n-2)\cdots (n-k+1)}{k!}. \]

Thus, we have that
\[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \cdots + \frac{n(n-1)\cdots (n-k+1)}{k!} x^k + \cdots \]

The radius of convergence \(R\) can be computed by using the following formula:
\[ R = \lim_{n \to \infty} \frac{c_n}{c_{n+1}}. \]

In this case when \(c_k = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}\), and so
\[ R = \lim_{k \to \infty} \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!} \div \frac{n(n-1)(n-2)\cdots(n-k)}{(k+1)!} = 1. \]

Therefore,
\[ (1+x)^n = 1+ nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots + \frac{n(n-1)(n-2)\cdots (n-k+1)}{k!} x^k + \cdots \]
holds whenever \(-1 < x < 1\).