So that definition would give the 'average rate of change of P(x)' i.e.
change in profit/change in quantity
For average profit which is the same as any average (arithmetic mean) i.e.
total/quantity
OK well I am working with Probability and random processes By Geoffrey Grimmett, David Stirzaker, the google books preview covers the section i am dealing with (1.1 and 1.2 right at the start). This uses the definitions given in the exercise...
For 3 we need the union of any two intervals to be in G
For the sigma field property we want all possible unions of all possible intervals to be in G
Point 3 basically requires [0,x) be in G for x<=1, which we do have.
In order for that sigma to hold we would need the interval (0,1) to be in...
Thanks yes! OK so we say
A=[ai,bi)U[a(i+1),b(i+1)U...U[ar,br)
Ac = omega/A
Then 3 is now easy
A=[ai,bi)U[a(i+1),b(i+1)U...U[ar,br)
B=[aj,bj)U[a(j+1),b(j+1)U...U[as,bs)
so AUB = [ai,bi)U[a(i+1),b(i+1)U...U[ar,br)U[aj,bj)U[a(j+1),b(j+1)U...U[as,bs)
AUB=[ak,bk)U[a(k+1),b(k+1)U...U[at,bt)...
Homework Statement
Let \Omega = [0,1)
Let G be the collection of all subsets of \Omega of the form
[a1,b1),\cup[a2,b2),\cup...\cup[ar,br)
For r any non-negative integer and 0<=a1
and a1 <=b1 <= a2 ....
Show that G is a field
Show that G is not a \sigma-field
Homework...
You can move through time.
To the extent that if you travel fast enough, you could return to find the earth extinct, or perhaps even the entire universe. Travelling at light speed you would experience zero time and so presumably you can travel to the end of the universe!
I don't study physics...
So to conclude
By the standard comparison test:
since (1-1/n)^3 >= 1/2 for n>=2
and n\sqrt{1+n^{-7}+2n^{-8} <= 2n
we must have
\frac{(1-1/n)^{3}}{n\sqrt{1+n^{-7}+2n^{-8}}}\geq\frac{1/2}{2n}=\frac{1}{4n}\]
More simply by the limit comparison test...
I do know the limit comparison however i just get the sequence tending to 0 rather than a finite L. So doesn't seem to be of use
Comparison test should work for 1/4n, for n>=2
\frac{(1-1/n)^{3}}{n\sqrt{1+n^{-7}+2n^{-8}}}\geq\frac{1}{4n}\]
So by comparison test the series diverges?