How to prove that this series bounded and monotonic

1. Dec 10, 2008

transgalactic

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?

2. Dec 10, 2008

Office_Shredder

Staff Emeritus
You need to show that $$x_n > x_{n+1}$$ for all n given that $$x_n(1-2^{-n}) = x_{n+1}$$

It should be fairly easy from here

3. Dec 10, 2008

transgalactic

that proves that its monotonic
how to prove that its bounded?

4. Dec 10, 2008

Office_Shredder

Staff Emeritus
Is it increasing or decreasing?

5. Dec 10, 2008

transgalactic

each next member is bigger then the previous one
so its increasing

1-1/2 1-1/4 1-1/8 etc..

6. Dec 10, 2008

Office_Shredder

Staff Emeritus
That's now what the sequence is. The sequence is

1/2, 1/2*3/4, 1/2*3/4*7/8 etc.

you should be able to see this from how xn is defined.

7. Dec 10, 2008

transgalactic

ok so it getting smaller and smaller
how to prove that its bounded?

8. Dec 10, 2008

HallsofIvy

Staff Emeritus
There's a pretty obvious lower bound. And since it is decreasing, isn't x1= 1/2 an upper bound?