[tex]\displaystyle\sum_{n\ge1}\frac{6^n(x+6)^n}{\sqrt n}[/tex]
By the ratio test, the series converges when
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{6^{n+1}(x+6)^{n+1}}{\sqrt{n+1}}}{\frac{6^n(x+6)^n}{\sqrt n}}\right|<1[/tex]
The limit is
[tex]\displaystyle6|x+6|\lim_{n\to\infty}\frac{\sqrt n}{\sqrt{n+1}}=6|x+6|[/tex]
which means the series converges when [tex]6|x+6|<1[/tex], or [tex]-\dfrac{37}6<x<-\dfrac{35}6[/tex].
