Recent content by Millennial

Recently, I have came across an interesting game that can be played fairly easily between two people. Here is how it is played: Both you and your opponent pick a four digit integer with no digit repetition. Then, you perform specific "guesses" so as to determine what their number is. A guess is...

It can be a whole bunch of other stuff, not just 0 and 1.

What is your definition of sequence, exactly?

We need the proof for this in elementary terms. If this is proven, the problem is basically solved. Edit: I just saw that the x_i had to be positive. Then, the proof is obviously very easy. The sum \displaystyle \sum_{i=1}^n i^2/n^2 has closed form \displaystyle \frac{(n+1)(2n+1)}{6n}. The...

Really? The infimum on the range of this function with the given constraint must be quite obvious.

Sorry, I misread the question. Using Wolfram, I can see that the maximum is reached in an equilateral triangle. However, I don't have an idea on why this is so. Using Lagrange multipliers, it is not hard to see that this is a critical point, but one needs to evaluate the other critical points...

From the Law of Sines, \displaystyle \frac{\sin(A)}{A} = \frac{\sin(B)}{B} = \frac{\sin(C)}{C}. Does that help?

ca = ac, so yes.

Another way to tackle the integral could be to somehow create the derivative of the base for the exponent in the denominator in the numerator. Then, you could use substitution.

The "number" you are speaking of is nonsensical. You can't have an infinite string of 5's, and then have numbers come after that. If there are infinitely many 5's, you can't add anything after that; because it would mean the string of 5's eventually terminates at some point, which violates the...

You could say d \propto r^{-1}, or more accurately d \propto v^{-1}.

Nope, the solution is completely false. The integral over that contour isn't 2\pi/3, it is -2\pi/3. The residues of the function: http://www.wolframalpha.com/input/?i=residues+of+the+function+sqrt%28z%29%2F%281%2Bz%5E3%29

Integration is obviously related to differentiation, both definite and indefinite ones. However, the inverse of differentiation is indefinite integration. Definite integration is an infinite sum of infinitely small things that just so happens to be computed using antiderivatives, see the...

It is solvable the way I gave without any general knowledge. Just substitute r_a = 1-r_b or vice versa after arrangement.

You want to maximize \pi(r_a^2 + r_b^2) based on the constraint that r_a + r_b = 1. Try starting with the identity (r_a+r_b)^2 = r_a^2 + r_b^2 + 2r_ar_b.