Calculating the limit of a function series

[akleron]

Hello every one !
I am trying to calculate the limit function of the following function series (and to determind if the series converge )
The only idea came to my mind is L'hopital's rule but i am not quite sure how to use it here because of the sigma..
Any help would be appreciated !
Thank you!

 

[skeeter]

haven’t worked with one of these in a long while, but the series looks geometric with [MATH]0 \le \dfrac{\ln{x}}{x} < 1[/MATH]
I may be out to lunch with that observation, so take it with a grain of salt.

 

[Steven G]

Yes, keeping x constant, it is geometric.

 

[akleron]

I am not sure I got it.
As the power goes to infinity the value inside sigma goes to zero.
But what is the sum of sigma ?

 

[skeeter]

if [MATH]|a| <1[/MATH], then [MATH]\sum_{k=1}^\infty a^k = \dfrac{a_1}{1-a}[/MATH]

 

[pka]

As the power goes to infinity the value inside sigma goes to zero.
But what is the sum of sigma ?

Look st this link. You see that \(\dfrac{\log(x)}{x}\le \dfrac{1}{e}\)