How to show that $1 \over \sqrt{1 - 4x} $ generates $\sum_{n=0}^\infty \binom{2n}{n}x^n $ I need to find the values of $a$ and $n$ in the following
$$ (1 + ax)^n  = \sum_{i=0}^\infty \binom{2i}{i}x^i$$
How can I compare things and show that $a=-4$ and $n = -{1 \over 2}$. It's easy to expand $1 \over \sqrt{1 -4x}$ see that it's valid but I'm looking other way around.
 A: At first, consider the series $\sum_{n=0}^{\infty}C_{2n}^{n}x^{n}$, as $\frac{C_{2n+2}^{n+1}}{C_{2n}^{n}}=\frac{4(2n+1)}{2n+2}$, we can judge that the series convergent in $[-\frac{1}{4},\frac{1}{4})$.
Note $f(x)=\sum_{n=0}^{\infty}C_{2n}^{n}x^{n}=\sum_{n=0}^{\infty}\frac{(2n)!}{n!n!}x^{n}$, then $f(0)=1$,
$$f^{\prime}(x)
=\sum_{n=1}^{\infty}\frac{(2n)!}{n!(n-1)!}x^{n-1}
=\sum_{n=0}^{\infty}\frac{(2n+2)!}{(n+1)!n!}x^{n}
=\sum_{n=0}^{\infty}\frac{(4n+2)(2n)!}{n!n!}x^{n}$$
$$xf^{\prime}(x)
=\sum_{n=1}^{\infty}\frac{(2n)!}{n!(n-1)!}x^{n}
=\sum_{n=0}^{\infty}\frac{n(2n)!}{n!n!}x^{n}$$
$$(1-4x)f^{\prime}(x)=2\sum_{n=0}^{\infty}\frac{(2n)!}{n!n!}x^{n}=2f(x)$$
then solve the last ordinary differential equation, we get $f(x)=\frac{c}{\sqrt{1-4x}}$, as $f(0)=1$, so $f(x)=\frac{1}{\sqrt{1-4x}}$, i.e. $\sum_{n=0}^{\infty}C_{2n}^{n}x^{n}=\frac{1}{\sqrt{1-4x}}, x\in[-\frac{1}{4},\frac{1}{4})$.
A: $$(1+ax)^{n}=\sum_{n=0}^{\infty}C_{2n}^{n}x^{n}=\sum_{n=0}^{\infty}\frac{(2n)!}{n!n!}x^{n}$$
Derivated at both sides, we get the first equation:
$$na(1+ax)^{n-1}=\sum_{n=1}^{\infty}\frac{n(2n)!}{n!n!}x^{n-1}$$
Derivated at both sides of the first equation, we get the second equation:
$$n(n-1)a^{2}(1+ax)^{n-2}=\sum_{n=2}^{\infty}\frac{n(n-1)(2n)!}{n!n!}x^{n-2}$$
Let $x=0$ in above two equations, we can get that:
$$na=2\,\,\text{and}\,\,n(n-1)a^{2}=12$$
then solve the equation and we can get that $a=-4$ and $n=-\frac{1}{2}$.
A: You can show that the sum is the solution of a differerential equation (try to express the derivative in function of the sum itself).
