<s>[INST] 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. 
 [/INST] $$\sum_{n=1}^\infty\dfrac{5^n+(-1)^n}{n^3}(x-2)^n=\sum_{n=1}^\infty\dfrac{\{5(x-2)\}^n}{n^3}+\sum_{n=1}^\infty\dfrac{\{(-1)(x-2)\}^n}{n^3}$$
Can you apply Ratio Test separately?
OR
$$\lim_{n\to\infty} \bigg|\frac{5^{n+1} + (-1)^{n+1}}{5^{n} + (-1)^{n}} \left(\frac{n}{n+1}\right)^{3} (x-2)\bigg|$$
$$=\lim_{n\to\infty}\left|(x-2)\left(\dfrac1{1+\dfrac1n}\right)^3\cdot\dfrac{5-\left(\dfrac15\right)^n}{1+\left(\dfrac15\right)^n}\right|=|5(x-2)|$$
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