Setting...
... where the sequence is defined as...
,
... after easy steps you obtain...
(1)
From (1) it derives that...
... so that...
Kind regards
hi,
i need to find the limit of the following sequence
lim (an)1/n
n®¥
where an the sequence defined recursively as follows
a1=1 and an+1=n(1-lnan) where ln is the natural logatithm
I can prove (using induction) that the sequence is increasing and not upper bounded but I don't know how to prove that. I'm sorry but I don't know how to write the sequence in the correct form, so I attached a doc file
lim (an)1/n=1 n®¥
In that case we have to find...
(1)
... where...
(2)
... and the sequence is defined as...
, (3)
In this case may help a preliminary test in order to undestand the behavior of the term in (2). Value of and computed till to n=1000 are plotted in figure...
Setting...
(4)
... il seems that the sequence is upper bounded, that means that...
(5)
... where . In this case is...
(6)
... so that...
(7)
All that however is not yet demonstrated ...
Here there are some value of with n increasing...
Kind regards