Firstly lets find the limit by solving
gives x = 4, so limit is 4
lets rewrite
we can see that if then and also that
so we can conclude that the sequence decreases and has a lower bound of 4.
hope this helps and makes sense!
cheers Nobby
a) X1=8 and Xn+1=.5Xn +2 for n within naturals. Show that Xn is bounded and monotone and find the limit?
b) X1> and Xn+1=2-1/Xn for n within naturals. Show that Xn is bounded and monotone and find the limit
for part a i can say x1>x2, thus the sequence is decreasing.
here's the outline to a more straightforward induction proof. i leave it to you to formalize it. we use induction to show that the sequence is monotonically decreasing (that is for all ), and hence is bounded above by the first term, namely 8
Note that so that
Now assume
then
adding 2 to both sides we get
but that means