Power Series Ratio Test

Question

Last time ive checked the ratio test was limit to infinity of a+1 / a

However, Ive approached a good amount of question that uses the inverse of that formula such as the following given below. Anyone want to explain why they inverted it?

Answer 1

It's just an arbitrary choice.

$\frac{|a_{n+1}|}{|a_n|}$ converges to something smaller than $1$ exactly when $\frac{|a_n|}{|a_{n+1}|}$ converges to something larger than $1$ (or to +infinity).

So they are really the same test. Sometimes one of the limits is slightly easier to calculate than the other one, but which one it is varies.

Answer 2

It is not ratio test. Here you are using ratio test however, that's not all. R is the radius of convergence. So let's say the series is $c_k(x-a)^k$ After you perform ratio test, you get $\lim {c_{n+1}\over c_n}(x-a)$. In order for the series to converge, this must be less than 1, so you get the invert. So basically it comes from the definition of radius of convergence.