Convergence of a sum for which x?

[stukbv]

Homework Statement

Consider the infinite series (1/n) * (xⁿ) where x is a real noumber. Find all numbers x such that
i) the series converges,
ii) series converges absolutely
iii) diverges to + infinity
iiii) does not converge.




2. The attempt at a solution
For this, i know it does not converge for x = 1 and for x<-1 , and it does converge absolutely for x = -1, i also know it will diverge to $$\infty$$ for x>1 but i am very unsure as to what happens for x between 1 and - 1 ?

Thanks a lot

 

[Dick]

What does the ratio test tell you?

 

[stukbv]

Ok, I see, so now i have that it converges for |x|<1 and the same for absolute convergence.
It does not converge at x=1 , x=-1 pr x<-1
and it diverges to infinity for x>1.
It this closer?

 

[Dick]

Ok, I see, so now i have that it converges for |x|<1 and the same for absolute convergence.
It does not converge at x=1 , x=-1 pr x<-1
and it diverges to infinity for x>1.
It this closer?

Closer. It converges for |x|<1 and diverges for |x|>1. The only points you have to worry about are x=1 and x=(-1). I'll agree it diverges at x=1. Can you say why? I don't agree that it diverges at x=(-1).

 

[HallsofIvy]

For x= -1, this is an alternating series and 1/n goes to 0.

 

[stukbv]

Oh yes, of course!
Thanks!