Question

Find the interval of convergence of the series (6^n (x+6)^n)/ sqrt(n)

Answer 1

`\displaystyle\sum_(n\ge1)(6^n(x+6)^n)/(\sqrt n)`

By the ratio test, the series converges when


`\displaystyle\lim_(n\to\infty)\left|((6^(n+1)(x+6)^(n+1))/(√(n+1)))/((6^n(x+6)^n)/(\sqrt n))\right|<1`

The limit is


`\displaystyle6|x+6|\lim_(n\to\infty)(\sqrt n)/(√(n+1))=6|x+6|`

which means the series converges when
`6|x+6|<1`, or
`-\frac{37}6<x<-\frac{35}6`.