Find the radius of convergence of $\sum_{n=1}^{\infty} \left(\frac{5^{n} + (-1)^{n}}{n^{3}}\right)(x-2)^{n}$ Find the radius of convergence of the following:
$$\sum_{n=1}^{\infty} \left(\frac{5^{n} + (-1)^{n}}{n^{3}}\right)(x-2)^{n}$$
My attempt: 
I used Ratio Test and managed to get until
$$\lim_{n\to\infty} \bigg|\frac{5^{n+1} + (-1)^{n+1}}{5^{n} + (-1)^{n}} \left(\frac{n}{n+1}\right)^{3} (x-2)\bigg|$$
I need to use L'Hopital Rule to get the answer. 

L'Hospital's rule is not necessary here, one may just write, as $n \to \infty$,
$$
\left|\frac{5^{n+1} + (-1)^{n+1}}{5^{n} + (-1)^{n}} \cdot\left(\frac{n}{n+1}\right)^{3} (x-2)\right|=5\left|\frac{1 + \frac{(-1)^{n+1}}{5^{n+1}}}{1 + \frac{(-1)^{n}}{5^{n}}} \cdot\frac{1}{\left(1+\frac1n\right)^{3}} \right|\cdot|x-2| \to \color{red}{5|x-2|}.
$$
