Radius of Convergence for Σ6n(x-5)n(n+1)/(n+11) Series | Solve for x

[ReidMerrill]

Homework Statement

Find all values of x such that the given series would converge

Σ6ⁿ(x-5)ⁿ(n+1)/(n+11)

Homework Equations

The Attempt at a Solution

By doing the ratio test I found that
lim 6ⁿ(x-5)ⁿ(n+1)/(n+11) * (n+12)/[6ⁿ⁺¹(x-5)ⁿ⁺¹(n+2)]
n→inf

equals 1/(6(x-5)) * lim (n+12)(n+1)/(n+11)(n+2)
This limit = 1 so to solve for the x I set
-1<1/6(x-5) and 1/6(x-5)<1 and found the (31/6)<x<(29/6)
but apparently this is incorrect. What am I doing wrong?

 

[Tom MS]

Homework Statement

Find all values of x such that the given series would converge

Σ6ⁿ(x-5)ⁿ(n+1)/(n+11)

Homework Equations

The Attempt at a Solution

By doing the ratio test I found that
lim 6ⁿ(x-5)ⁿ(n+1)/(n+11) * (n+12)/[6ⁿ⁺¹(x-5)ⁿ⁺¹(n+2)]
n→inf

Here it looks like you flipped the wrong fraction.
Ratio test is more like \lim_{n\rightarrow \infty} {\frac{f(n+1)}{f(n)}}
Where f is the function under the sigma. In my understanding, this would flip the fraction the other way from what you have.