- #1

chwala

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- Homework Statement
- Determine the convergence or divergence of the sequence ##a_n= \left[\dfrac {\ln (n)^2}{n}\right]##

- Relevant Equations
- convergence knowledge

##a_n= \left[\dfrac {\ln (n)^2}{n}\right]##

We may consider a function of a real variable. This is my approach;

##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]##

Applying L'Hopital's rule we shall have;

##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[ \dfrac {2}{x}\right]=0##

because ##f(n)=a_n## for every positive integer ##n##, then we may conclude that the sequence converges to ##0##.

I would appreciate any insight on this...cheers.

We may consider a function of a real variable. This is my approach;

##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]##

Applying L'Hopital's rule we shall have;

##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[ \dfrac {2}{x}\right]=0##

because ##f(n)=a_n## for every positive integer ##n##, then we may conclude that the sequence converges to ##0##.

I would appreciate any insight on this...cheers.

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