## SolomonZelman one year ago I have a question about an understanding of a concept. It is easy - not hard.

1. SolomonZelman

Say, I got any series $$\large\color{black}{ \displaystyle \sum_{{\rm n}=k}^{\infty}{\rm A}_{\rm n} }$$ where the root test for converges in practical, for instance (in red) $$\large\color{black}{ \displaystyle \sum_{{\rm n}=k}^{\infty}\frac{x^{\rm n}}{(5{\rm n }+3)^{\rm n}} }$$ (or if you can give a better example, please do) why does the nth root test show that series converges if the (absolute value of the) result is less than 1?

2. SolomonZelman

the $$\large\color{black}{ \displaystyle \lim_{ n\rightarrow \infty }~\sqrt[n]{A_n}}$$ is the geometric ratio - simply speaking. It is expolring how the series behaves as we get to so to speak "infinitiith" terms.

3. SolomonZelman

For example $$\large\color{black}{ \displaystyle \lim_{ n\rightarrow \infty }~\sqrt[n]{\frac{\rm x^n}{\rm (5n+3)^n}}}$$ $$\large\color{black}{ \displaystyle \lim_{ n\rightarrow \infty }~\frac{\rm x}{\rm 5n+3}=r}$$

4. SolomonZelman

and if that expression $$\large\color{black}{ \displaystyle \lim_{ n\rightarrow \infty }~\frac{\rm x}{\rm 5n+3}}$$ has an absolute value (forgot absolute value of the limit) then it converges, just as any elementary geom. series would with |r|<1

5. SolomonZelman

tnx for any views, and out:)