Answer
We prove the following:
Case 1. $a$ is not a whole number
The radius of convergence is $R=1$.
Case 2. $a$ is a positive whole number
The radius of convergence is $R = \infty $.
Work Step by Step
We have the binomial series: ${T_a}\left( x \right) = \mathop \sum \limits_{n = 0}^\infty \left( {\begin{array}{*{20}{c}}
a\\
n
\end{array}} \right){x^n}$.
Write out the expansion:
${T_a}\left( x \right) = \mathop \sum \limits_{n = 0}^\infty \left( {\begin{array}{*{20}{c}}
a\\
n
\end{array}} \right){x^n} = 1 + ax + \dfrac{{a\left( {a - 1} \right)}}{{2!}}{x^2} + \dfrac{{a\left( {a - 1} \right)\left( {a - 2} \right)}}{{3!}}{x^3} + \cdot\cdot\cdot + \dfrac{{a\left( {a - 1} \right)\left( {a - 2} \right)\cdot\cdot\cdot\left( {a - n + 1} \right)}}{{n!}}{x^n} + \cdot\cdot\cdot$
Case 1. $a$ is not a whole number
We examine the radius of convergence by computing the ratio and its limit with ${a_n} = \dfrac{{a\left( {a - 1} \right)\left( {a - 2} \right)\cdot\cdot\cdot\left( {a - n + 1} \right)}}{{n!}}{x^n}$.
$\rho = \mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{{{a_{n + 1}}}}{{{a_n}}}} \right|$
$\rho = \mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{{a\left( {a - 1} \right)\left( {a - 2} \right)\cdot\cdot\cdot\left( {a - n} \right)}}{{\left( {n + 1} \right)!}}{x^{n + 1}}\cdot\dfrac{{n!}}{{a\left( {a - 1} \right)\left( {a - 2} \right)\cdot\cdot\cdot\left( {a - n + 1} \right)}}\cdot\dfrac{1}{{{x^n}}}} \right|$
$\rho = \left| x \right|\mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{{a - n}}{{n + 1}}} \right|$
$\rho = \left| x \right|\mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{a}{{n + 1}} - \dfrac{n}{{n + 1}}} \right| = \left| x \right|\mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{a}{{n + 1}} - \dfrac{1}{{1 + \dfrac{1}{n}}}} \right|$
$\rho = \left| x \right|$
We find that $\rho \lt 1$ if $\left| x \right| \lt 1$. By the Ratio Test, the binomial series is convergent if $\left| x \right| \lt 1$. Thus, the radius of convergence is $R=1$.
Case 2. $a$ is a positive whole number
In this case, $\left( {\begin{array}{*{20}{c}}
a\\
n
\end{array}} \right)$ is zero for $n \gt a$. Thus, the series breaks off at degree $n$. This implies that ${T_a}\left( x \right)$ is a polynomial of degree $a$.
Since ${T_a}\left( x \right)$ is a polynomial, it converges for all $x$. Thus, the radius of convergence is $R = \infty $.
