<s>[INST] Interval of convergence of $\sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!}$ 
Given the series 
$$ \sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!} \quad \quad k \geq 1 $$
  Find the interval of convergence.

I started by applying the Ratio test
$$
\lim_{n\to \infty}\left|\frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)(k+n)x^{n+1}}{(n+1)!}\cdot \frac{n!}{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}\right|$$
$$\lim_{n\to \infty}\left|\frac{(k+n)x}{(n+1)}\right|$$
to show that the series converges when $|x| \lt 1$.
However, when I test the end points of $(-1,1)$ for convergence, I end up with two series whose convergence I am unable to show. Namely,
$$
\sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)}{n!} 
$$
and 
$$
\sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)(-1)^n}{n!} 
$$
How can I show that these two series converge or diverge?
 [/INST] Note that for $k\ge1$, we have
$$
\frac{k(k+1)\cdots(k+n-1)}{n!}=\frac k1\frac{k+1}2\cdots\frac{k+n-1}{n}\ge1
$$
Thus, for $|x|=1$, the terms do not go to $0$.
</s>