There are several ways to check whether an integer is prime. The usual one is to simply to check for divisibility by every prime number up to the square root of your number, as you suggested. This is not the only one though. For large suspected primes you usually use algorithms which rely on...
If you have no idea how to approach the problem I would suggest the following process:
Start listing the A's that could work in this problem, i.e. the A has 8 as a divisor and which itself is a divisor of 192. For instance your list might start:
A=8
A=16
...
For every such A ask yourself what B...
Yes (slight lie, see below). Every log is defined precisely on the positive numbers.
This is a slight lie because in complex analysis we actually extend log and define it for many complex numbers. However we will never be able to include 0 in the domain (without making it discontinuous), and...
The part you seem to forget is "some point c of its domain". 0 is not in the domain of ln so this is why your observation is not a problem. The domain of ln is the positive numbers.
It is however an interesting observation in its own right and it implies that you cannot possibly define ln(0) in...
Yes every point of E is either an isolated point or a limit point of E.
The proof is not complicated and you should consider it an exercise. It is the kind of statement you should be comfortable proving after reading this section of Baby Rudin.
Look at the left hand of your last equation. We have:
n(n-1)\cdots (n-r +1)
times
(n-r)! = (n-r)(n-r-1)\cdots (2)(1)
Multiplying these together you get
n(n-1)\cdots (n-r +1)(n-r)\cdots (2)(1) = n!
So you want x->0 and \beta \to 0 right?
Your series looks a lot like a geometric series. In fact
\sum_{i=1}^\infty \beta^nx^{3+2n} = x^3\sum_{i=1}^\infty (\beta x^2)^n
For small enough \beta and x you should get a nice closed form from which you can more easily see the series' asymptotic behavior.
What exactly do you require of your function f? I'm assuming it is real-valued, but what set is it from? You example of log is just a function \mathbb{R}^+ \to \mathbb{R}. Are we just looking for a function A \to \mathbb{R} where A is any set of real numbers that contains a and is closed under...
As I have come across the term in these contexts the modulus isn't just "some" real-valued function but rather the function
\|z\| = z\overline{z}
where \overline{z} denotes the conjugate, so:
\overline{a} = a\qquad\textrm{for real a}
\overline{a+bi} = a-bi\qquad\textrm{for complex a+bi}...
But then it doesn't matter. I only claimed that every proof will eventually appear. I did not claim that for every true theorem we would eventually find a proof.
I now realize I also said "and every theorem" which of course is incorrect (or at least misleading). What I meant was "and every...
What do you mean?
A proof can be written as a string of symbols chosen from a finite alphabet. There are finitely many strings of n symbols for a fixed integer n, so a computer will be able to go through all combinations of length n in some time T(n). It would then take a computer.
T(1)+T(2)...
For theorems it's actually a lot like art. Most theorems are worthless and the computer won't be able to rule out garbage theorems.
If you already have a statement you want to find a proof for, then it could be of some use, but it's still of limited use for the following reasons:
1) If the...
Yes. At any given point in time it will only have given us finitely many theorems, but there are infinitely many (assuming a moderately interesting axiom system and a computer with finite computing powers).
What would be true is that every theorem and proof would eventually appear, but you will...
As Tedjn mentioned you need to restrict cosine to a small enough interval due to the behavior near pi/2 (and kpi+pi/2 for all integers k). In fact it's a well-known trigonometric identity that \sin(h)/h \to 1 as h\to 0 so this gives us:
\left|\frac{\cos(\pi/2-h)...
I have used their books in high school to self-study for math competitions (haven't seen their calculus and pre-calculus books as they are recent additions). I liked them and feel they adequately prepared me for the earlier stages of competition, and later on prepared me for the more advanced...
Disclaimer: I have no first-hand knowledge of grad school admission as I'm myself an undergrad math student, but since you haven't gotten many responses I'm just going to pass on some of the advice given to me.
From what I've heard hobbies matter VERY little to grad school admission. Unless...
People don't think of it as a bottom up hierarchy. If you've already shown that (1) and (2) are equivalent then you can just choose with whatever you find more convenient at the time. For many mathematical objects there are many definitions, and you just pick whatever suits your problem...
There are plenty of ways to show it.
For simplicity let z_n = |x_n-y_n| (no need to keep track of both x_j and y_j).
Method 1 (means):
The generalized mean inequality says that:
M_p(a_1,a_2,\ldots,a_n) = \left(\frac{a_1^p + a_2^p + \cdots + a_n^p}{n}\right)^{1/p}
Is increasing as a function...
This is a special case of the Cauchy–Schwarz inequality which in the finite-dimensional real case state:
\sum_{i=1}^n a_i b_i \leq \left( \sum_{i=1}^n a_i^2 \right)^{1/2} \left( \sum_{i=1}^n b_i^2 \right)^{1/2}
For all real numbers a_i, b_i.
Well if you really want a numeric approximation I just asked Maple to compute 250 digits of this. I'm not sure exactly how Maple does its floating-point computation, but I suspect the long string of 0's at the end is a sign that it doesn't handle such high-precision numbers by default (but up to...
Let a = 1.0967*10^{-86}
Square the equation to get it to get
\frac{1-x^2}{x^2} =a^2
Now solve for x^2 and take the positive square root (x must be positive since a is positive).
x is going to be VERY close to 1 (in fact I expect most software will report x=1 if you ask for a numerical...
arxiv is a great resource, but the fact that a paper made it onto arxiv does not mean that the paper is good or even correct. Plenty of crackpots manage to get their papers on arxiv.
Arxiv does not try to fill the same role as a respectable peer-reviewed journal and you shouldn't consider it...
What people say when they say that an equation of degree n has n roots is that given a polynomial P(x) of degree n, it has n complex roots (where n is a non-negative integer). However x^0=1 so the polynomial in your last example is:
P(x) = x^0 - 1 = 1-1 = 0
This does not have degree 0. We...
Valid according to what standards?
If you're wondering whether it's actually easier to calculate the right hand side, then the answer is yes assuming you are not completely naive (naive approach: calculate sqrt(x), then apply ceil or floor).
To calculate \lfloor \sqrt{x}\rfloor simply...
Yes this would work just as well. I used my approach mainly because it also gives you the actual limit (whether you care for it or not), and it uses more elementary tools.
I guess the limit comparison test is actually a bit easier to carry out now that you point it out and it may have been...
Well it's clearly equivalent to testing for convergence of:
\sum_{n=3}^\infty \frac{4}{n^2-4}
so let's do this instead (the constants turn out nicer). We note:
\frac{4}{n^2-4} = \frac{1}{n-2}-\frac{1}{n+2}
You can use this and a standard telescopping argument to show that it converges.
From the hint you know that you can write the polynomial x^3 as:
x^3 = a_0\binom{x}{0} + a_1\binom{x}{1} + a_2\binom{x}{2} + a_3\binom{x}{3}
for constants a_0,\ldots,a_3. By substituting appropriate values for x you should be able to work out these constants. By plugging this expression into...
The last line here is wrong. The appropriate identity states:
\cos(\alpha-\beta) = \cos\alpha\,\cos\beta + \sin \alpha\,\sin\beta
So you should have gotten:
RHS=xcos\frac{\pi}{2}+sin(cos^{-1}x)sin\frac{\pi}{2}
In the future a good way to identify an error in an argument about trigonometric...