Show that the sequences defined by:
satisfies and is decreasing. Deduce that the sequence is convergent and find it's limit.
(we will also require the other root later so we may as well anounce it here)
Now suppose , then let , and we have:
But , so:
and as , we have , and so .
Hence as , for all .
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 .