Show that the sequences defined by:
;
satisfies and is decreasing. Deduce that the sequence is convergent and find it's limit.
Put then is a root of .
(we will also require the other root later so we may as well anounce it here)
Now suppose , then let , and we have:
So:
But , so:
and as , we have , and so .
Hence as , for all .
Now consider:
This is greater than for between the roots of that is between .
That is is decreasing while and and as , is a decreasing sequence and also bounded below, so converges.
Now if converges the limit satisfies:
which has roots and But it cannot converge on as the sequence is decreasing so the limit is .
RonL