Value function iteration
It is not easy for us to calculate intercept term for value function.
We know the following
for j=1,2,3,. . .
from one of the results yesterday. And it is obvious that
for
j=1,2,3,. . .
Now we consider the intercept part only in each j.
Removing 
from each value
function, we get
When j=0
When j=1
When j=2
When j=3
.
.
.
When j=n+1
Since 
for 0<a<1
when we consider infinitely many terms
That is
Here, we can regard 
and it converges
so we rewrite it as
Thus
Thatfs what we want in value function 
while
calculated yesterday.