Konverge III

Let $a_n = \dfrac{(n!)^k}{(nk)!} x^n$ . The Ratio Test shows us that the series will converge if $\lim_{n \to \infty} \left | \dfrac{a_{n+1}}{a_n} \right | = \lim_{n \to \infty} \left | \dfrac{\dfrac{(n+1)!^k x^{n+1}}{(nk+k)!}}{\dfrac{(n!)^k x^n}{(nk)!}} \right | = \lim_{n \to \infty} \left | \dfrac{(n+1)^k}{(kn+1)(kn+2)(\cdots)(kn+k)} \cdot x \right | < 1 \Leftrightarrow$ $\Leftrightarrow \lim_{n \to \infty} \dfrac{n^k \cdot \left (1+\frac{1}{n} \right )^k \cdot \left |x \right | }{n^k \cdot \left (k+\frac{1}{n} \right ) \cdot \left (k+\frac{2}{n} \right ) \cdot (\cdots) \cdot \left (k+\frac{k}{n} \right )} = \lim_{n \to \infty} \dfrac{\left (1+\frac{1}{n} \right )^k \cdot \left |x \right | }{k \cdot \left (1+\frac{1}{kn} \right ) \cdot k \left (1+\frac{2}{kn} \right ) \cdot (\cdots) \cdot k \left (1+\frac{1}{n} \right )} < 1 \Leftrightarrow$ $\Leftrightarrow \lim_{n \to \infty} \left ( \dfrac{1}{k^k} \cdot \left |x \right | \cdot \dfrac{\left (1+\frac{1}{n} \right )^k}{\left (1+\frac{1}{kn} \right ) \left (1+\frac{2}{kn} \right ) (\cdots) \left (1+\frac{1}{n} \right )} \right ) = \dfrac{1}{k^k} \cdot \left |x \right | \cdot \lim_{n \to \infty} \left ( \dfrac{\left (1+\frac{1}{n} \right )^k}{\left (1+\frac{1}{kn} \right ) \left (1+\frac{2}{kn} \right ) (\cdots) \left (1+\frac{1}{n} \right )} \right ) < 1 \Leftrightarrow$ $\Leftrightarrow \dfrac{\left |x \right |}{k^k} < 1 \Leftrightarrow \boxed{\left |x \right | < k^k.}$