How powerful are continued fraction representations?
From what I understand, they could be used to exactly represent some irrational numbers
So, could they represent any root of an nth degree polynomial equation?
Specially where n>4, since 5th degree roots are not guaranteed to have an...
I need to prove:
(n+1)*(log(n+1)-log(n) > 1 for all n > 0.
I have tried exponentiating it and I got
( (n+1)/n )^(n+1) < e.
And from there I couldn't go any farther, but I do know that it is true by just looking at its graph.
Could anybody help me please?
I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives.
I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0):
F(x+Δx) - F(x) = F'(x) * Δx
The Δx factor...
I've been taught that for any system of linear equations, it has a corresponding matrix.
Why do people sometimes use systems of linear equations to describe something and other times matrices?
Is it all just a way of writing things down faster or are there things you could do to matrices that...
Can somebody help me please, I've tried solving this for hours but I still couldn't get it.
Given that a, b, c, d are positive integers and a+b=c+d.
Prove that if a∗b < c∗d,
then a∗log(a)+b∗log(b) > c∗log(c)+d∗log(d)
How do I do it?
Given a sequence, how to check if it converges?
Assume the sequence is monotonic but the formula that created the sequence is unknown.
My first thought was if:
seq(n+2) - seq(n+1) < seq(n+1) - seq(n) , is always true as n->infinity then it is convergent.
Or in other words, if the difference...
I would want to change this expression to another one that would have P easily accessible.
\frac{\left \lceil (n+1)^{P} \right \rceil}{\left \lceil n^{P} \right \rceil}
As an example of 'accessible'; given (n+1)^P / n^P, I can make P more accessible by using log(). Then it becomes P *...
How do I solve for x in the relationship below:
nx ~ n ln(n), as n -> infinity
The answer that I'm getting is x=1, but that must be wrong since 1 ~ ln(n) as n-> infinity is wrong.
The sequence that I'm working with is sorta monotonic.
It's monotonic most of the time, sometimes there's one number that ruins the trend like 0.0001001 in
0.1, 0.01, 0.001, 0.0001001, 0.0001, ...
but those are very rare. Btw, the sequence above is just an example.
Is Richardson extrapolation...
People say that if you could break a function down into these three functions (constant, successor, projection or sometimes called initial/basic functions) using some operators, then it is primitive recursive.
What makes these three functions so special?
I've been looking at this proof -> http://basics.sjtu.edu.cn/~liguoqiang/teaching/comp14/materials/Ackermann.pdf [Broken]
At the bottom of the second page, there is this thing which looks like this:
http://s24.postimg.org/sxtz7pef9/what.png
Could someone please explain what that function with...
I have a sequence of natural numbers which is generated by a really complicated function.
This function only takes in natural numbers.
How do I solve for a function that is asymptotic to the sequence?
If I have a sequence of ans, I want a function that satisfies f(n) ~ an as n -> infinity.
Assume that I have absolutely no clue to what is the formula used to generate a sequence.
How do I know what kind of formula that is? (Exponential / Linear / Polynomial / etc)
Also assume that there is only 1 formula that generates the sequence.
I have read somewhere before that:
f'(x) ~...
Why is it significant enough to be fundamental?
Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.
public static int function(int n, int a, int b){
if (n>0){
if(b==0)
return a;
else
return function(n-1,a,function(n,a,b-1));
}
return b;
}
I have a recursive function of this form, how do I convert it to a loop?
The 2nd law of thermodynamics state that entropy increases with time and entropy is just a measure of how hard it is to distinguish a state from another state (information theoretical view) or how hard it is to find order within a system (thermodynamic view). There are many ways to view entropy...
The real projective line states that there is not difference between positive and negative infinity (maybe except the path needed to be taken to "reach" either one of them) and they are actually connected.
There are a lot of ways to get a definite sum from a divergent series; one of which is...
you can list and match up all rational numbers with irrational numbers this way..
lets say i have an irrational number 'c'.
Rational->Irrational
r1->cr1
r2->cr2
.
.
.
rn->crn
There exists an irrational number that is not on this matching, (not equal to any of the crx's)
this...
is it also possible to transform any these kinds summation to any product notation:
1. infinite - convergent
2. infinite - divergent
3. finite (but preserves the "description" of the sequence)
For example, I could describe the number 6, from the summation of i from i=0 until 3.
Could I...