Determine the radius of convergence of the following series $$C_n = (-1)^n\dfrac{m(m - 1)\cdots(m - n + 1)}{n!n!}C_0$$
I came to the conclusion that we can write $$C_{n+1} = C_n\dfrac{n - m}{(n+1)^2}$$ and then from the radius of convergence formula, we know that $$\displaystyle\lim_{n\to\infty}\left|\dfrac{C_{n + 1}}{C_n}\right| < 1$$ From here onwards, I don't know how to continue.
Any help would be appreciated.
Hint:
As you stated, you can use the Ratio Test: $$\left|\frac{C_{n+1}}{C_n}\right|=\left|\frac{{m \choose n+1}\cdot\frac{1}{(n+1)!}}{{m \choose n}\cdot\frac{1}{n!}}\right|=\left|\frac{m-n}{n+1}\cdot\frac1{n+1}\right|=\left|\frac{m-n}{(n+1)^2}\right|\to0, \quad n\to\infty.$$
In the comments you added that the series is $\sum c_k x^k$, therefore to find the radius of convergence just compute the limit $$\lim_{n\to\infty}\left|\frac{C_{n+1}\cdot x^{n+1}}{C_n\cdot x^{n}}\right|$$ using the hint above. Note that if the value of the limit above is $0$, then your radius of convergence is $\infty$.