Question

I have a question about an understanding of a concept. It is easy - not hard.

Answer 1

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?

Answer 2

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.

Answer 3

For example

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

\(\large\color{black}{ \displaystyle \lim_{ n\rightarrow \infty }~\frac{\rm x}{\rm 5n+3}=r}\)

Answer 4

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

Answer 5

tnx for any views, and out:)