I was wondering if i could get some help with this question:
Define a sequence by and for Use the Monotone Convergence Theorem to show that converges and find its limit.
Ok it is not too hard to prove by induction that . But I am having trouble proving the sequence is decreasing. I have
, but I have no guarantee that this is less than or equal to zero!
Any help would be appreciated!
Let's write the difference equation as...
(1)
... so that its a constant coefficients homogeneous linear difference equation, and its solution is on the form...
(2)
The correponding characteristic equation is...
(3)
... the solutions of which are and , so that is...
(4)
The 'initial condition' and give us and so that the solution of (1) is...
(5)
... and is also ...
Kind regards
ops! ... very sorry! ...
The difference equation is then...
(1)
... and the characteristic equation...
(2)
... the solution of which are , so that the solution of (1) is...
(3)
At this point it doesn't matter which are the 'initial conditions' because in any case is...
(4)
Kind regards