1. Homework Statement
Does 1/[n(log(n))^2] converge or diverge
2. Homework Equations
We know that Does 1/[n(log(n))] diverges by integral test
3. The Attempt at a Solution
1. Homework Statement
Why does the series 1/[nlog(n+1)] diverge
2. Homework Equations
We know that 1/[nlog(n)] diverges by the integral test. However the question as written does not lend itself to be any integral precisely.
3. The Attempt at a Solution
1. Homework Statement
Prove that the series:
1/[n log(n)]^1.1 converges
2. Homework Equations
3. The Attempt at a Solution
We know that nlogn is equal to d[log(log(n))] and use the integral test to show that it diverges.
However, I have no idea how to deal with the 1.1th power.
1. Homework Statement
How to show that sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges?
2. Homework Equations
3. The Attempt at a Solution
The above expression is asymptotically equivalent to 1/2n which diverges as the harmonic series diverges.
However, a rigorous proof is required...
1. Homework Statement
Does limit ln(n)/n^c -> 0 for any c>0?
2. Homework Equations
3. The Attempt at a Solution
I wonder if there is an
1.Epsilon Delta Proof
2.Proof using BigO SmallO notation.
Thanks
1. Homework Statement
Does the series n^2/(n^3+n^2) diverge?
2. Homework Equations
We know that 1/n diverges
3. The Attempt at a Solution
lim n^2/(n^3+n^2) =lim 1/n
Therefore intuitively it should diverge like 1/n
However, I am not very good at the Big O Small O...
1. Homework Statement
How to rigorously (real analysis) prove that for all real c>0
Exists N such that for all n>N
ln(n)<n^c
2. Homework Equations
3. The Attempt at a Solution
The fact can be shown using graphical calculator
Hi,
I wonder how to prove that ln(x) < sqrt(x) for all x>0?
Please enlighten me on two possible way to prove this .
Proof1. Using calculus and derivatives
Proof2. Since I'm taking real analysis, I wonder if it is possible to use taylor series to show this in an elegant way.
Hi,
I know I can use a graphical calculator to easily show that
How to prove ln(x) < sqrt(x) for all x>0
But I wonder if there is a rigorous way to demonstrate this.
1. Homework Statement
How to find all complex number Z such that Z^5=-32
2. Homework Equations
Euler equation
e^it=cost+isint
3. The Attempt at a Solution
I guess a naive way to solve is that since Z^5=(-2)^5
Therefore Z=-2, but this obviously too good to be true.
I have no...
1. Homework Statement
Hi, We know the famous central limit theorem for means.
I wonder if there is a central limit theorem for Median?
If so under what regularity condition, does the median converge to a normal distribution with mean and variance equal to what?
2. Homework Equations...
1. Homework Statement
Let X and Y be independent and normal, then we know that
It must be the case that X+Y and X are jointly normal
Therefore we can apply the projection theorem:
which states that if A and B are jointly normal then VAR(A|B)=VAR(B)-\rho^2VAR(B) , apply the theorem to A=X+Y...
What's the proof of this fundamental theorem?
Let (X,B,u) and (Y,C,v) be sigma finite measure spaces, Let f in L1(X,B,u) and g in L1(Y,C,v). Let h(x,y)=f(x)g(y).
Then, h is in L1(XxY,BxC,uxv) and
\int hd(u\times v)=\int fdu \int gdv
should be an easy application of fubini,but i really have no...
define F(x)=x, then uF is the lebesgue stiljes measure
duF=dx
Let y=x^2
we all know that how to transform \int f(y)dy into \int f(x^2)2xdx (***)
But how exactly would one use the transformation theorem ?
Ie. T be a measurable transformation from X to Y, u is a measure on X...
Suppose that A and B follow geometric brownian motion, where zA, and zB follow wiener process
dA/A=a*dt+b*dzA
dB/B=c*dt+d*dzB
dzA*dzB=e*dt
What stochastic process does A/B follow?
This is not a homework question(I am sure it's almost trivially easy to those who learned the stuff). I am very...
Prove (0,1) ~ [0,1]
I can think of an indirect proof:
1st step: make (0,1) ~ N , using a tangent function that is a 1-1 mapping from N to (0,1).
2nd step: since (0,1) is a subset of [0,1], if (0,1) is uncountable, then [0,1] must be uncountable
Problem: But these two steps doesn't...
Hi I've got a tough analysis proof. If I can do this on my own I might as well be Cantor himself.
The first step is :
Let C and N denote the collection of every cauchy sequence and null sequence (consisting of rationals) , prove that N is a subset of C
Second step is :
Prove N induces a...