Xn=(1-1/2)(1-1/4)..(1-(1/(2^n))
i tried to prove that its monotonic
by :
1-1/(2^n) = (2^n-1)/2^n
2^n -1 <2^n
obviously its correct
the numerator of each object is smaller then the denominator.
what now??
and how to prove that its bounded?
Xn=(1-1/2)(1-1/4)..(1-(1/(2^n))
i tried to prove that its monotonic
by :
1-1/(2^n) = (2^n-1)/2^n
2^n -1 <2^n
obviously its correct
the numerator of each object is smaller then the denominator.
what now??
and how to prove that its bounded?
Is there a reason you keep calling these series instead of products?
You want to prove monotonicity and boundedness of . Boundedness is simple
As for monotonicity you are making a mistake we are talking about the monotonicty of , that is NOT . So we need to show that . Since both sides are positive for all n we have that
. Simplifying gives which is obviously true for all positive n.