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
Is this one more difficult? How do we use the comparison test please to show the following?
sqrt[(n^8-10n^3+6)]/{sqrt[(n^7+100n^4+1)]*sqrt[n^3-500n^2+1]}
Please check my answer:
for the comparison test, claim that there exists N such that for all n>=N,
(n^4-10n^3+6)/(n^5+100n^4+999) >(n^4-10n^3+6)/(2n^5)
Proof,need the following
n^5+100n^4+999<2n^5
n+100+999/n^4 <2n
100+999/n^4 <n
Therefore N=101?
Sorry I changed my question to
(n^4-10n^3+6)/(n^5+100n^4)
There is no easy quintic formula so I made up this example to deter any easy way of explicity finding the N
Thanks for the reply. Here is a harder question:
(n^4-10n^3)/(n^5+100n^4)
It is not obvious that we can explicitly find an N such that for all n>N the above expression is larger than 1/n
My point is that is there any rigorous way to show that if lim an= lim bn
then series an diverges if and...
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.
t=180degree=-5pi,-3pi,-pi,pi,3pi,5pi,7pi,9pi,11pi?
Therefore -32=32*e^(it)=32*[cos(pi)+isin[pi]]
Sorry how do I use de Moivre's formula? Never taken complex analysis
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...