# Search results

1. ### Does 1/n(log(n))^2 converge or diverge

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
2. ### Why does 1/[nlog(n+1)] diverge

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
3. ### Why does 1/[n log(n)]^1.1 converge

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.
4. ### Sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges?

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...
5. ### Does limit ln(n)/n^c -> 0 for any c>0?

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
6. ### Does n^2/(n^3+n^2) diverge?

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...
7. ### Ln(n)<n^c for all c>0?

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
8. ### Prove ln(x) < sqrt(x) for all x>0

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.
9. ### How to prove ln(x) < sqrt(x) for all x>0

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.
10. ### How to find all complex Z such that Z^5=-32

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...
11. ### Is there a central limit theorem for Median?

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...
12. ### Sum of independent uniform distribution conditional on uniform

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...
13. ### Product measure

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...
14. ### Transformation of lebesgue integral

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...
15. ### Easy question on stochastic process

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...
16. ### (0,1) ~ [0,1]

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...
17. ### Cantor's definition of reals

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...