Dear all,
I'm having some trouble with the following limit:
n(x^n) as n tends to infinity, knowing that |x| < 1
I need it to calculate a series.
Please note that I am not allowed to use exp and ln as we haven't covered that in the syllabus yet. However, I'd be glad to know of a solution using these functions too.
Thanks in advance.
Yack
PS: this is my first post here, sorry if I left this message in the wrong place.
Theorem: Let be a sequence such that is a convergent sequence with limit . If then the sequence and if then is divergent.
Proof: Say (it cannot be that because this is a sequence of non-negative terms) and then for . This means, for , choose so small that (which is possible since ). Then it means by the Squeeze theorem because . Thus, . Q.E.D.
The case with , is similar, i.e. pick an so small that and then .