Recent content by txy

But the answer is 46/5 ...

The probability of the first k balls is red is \frac{C_{46}^{k}}{C_{50}^{k}} This is so difficult to sum!

Homework Statement There are 50 balls in a bag, all are the same except for the color. 4 of them are white while the rest (46) of them are red. Now randomly draw balls from the bag, one at a time, without replacement. What is the expected number of red balls that you will draw before you...

I think I've solved it. Let \overline{X}_k = \frac{1}{k}\left(x_1 + x_2 + \dotsb + x_k\right) Then x_1 + x_2 + \dotsb + x_k = k\overline{X}_k Because \overline{X}_k is the mean, we have \overline{X}_k \leq x_k \leq x_{k+1} \leq \dotsb \leq x_n So x_1 + x_2 + \dotsb + x_k + x_{k+1} +...

Homework Statement Given n real numbers x_1, x_2, \dotsb , x_n which satisfy 0 \leq x_1 \leq x_2 \leq \dotsb \leq x_n, show that \frac{x_1 + x_2 + \dotsb + x_k}{x_1 + x_2 + \dotsb + x_n} \leq \frac{k}{n}, \forall 1 \leq k \leq n. Homework Equations The Attempt at a Solution If x_1 = x_2 =...

Yes now I can prove it. Thanks!

But this question that I posed is in the Limits chapter of my textbook, which comes before the chapter on Differentiation. I wonder if there's another way to prove that inequality. Perhaps I could change it to the form x < e^x , but I still can't make any progress.

That's great. In either case, I need to use the fact that ln n < n for all n > 1. How do I prove this?

Homework Statement Find lim_{n \rightarrow \infty} \sqrt[n]{n ln(n)}. Do not use L'Hospital's Rule or Taylor Series. Homework Equations The Attempt at a Solution I suspect I need to set up some inequality for this and then apply Squeeze Theorem. But I can't find any inequality...

hmm how about you consider different cases? The simplest case is when the edges of the cube are parallel to the axes. Then it is simple to show that the length of one side is an integer. The second case is when 2 pairs of opposing edges (a total of 4 edges) are parallel to one of the...

Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...

there are theorems that state that when the function satisfies certain conditions, the Fourier series of the function converges to some expression. if you have learned these theorems, then it's quite easy to show that the Fourier series converges to the function itself.

why not take out 1/2 from the partial fraction decomposition so that it's more obvious? \sum^{+\infty}_{n=1}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4}) = \frac{1}{2}\sum^{+\infty}_{n=1}(\frac{1}{n}-\frac{2}{n+1}+\frac{1}{n+2}) =...

Oops I didn't phrase my question properly. I should have written "I'm not sure how to obtain the other acceleration expression." to avoid ambiguity. Thankfully you understood what I meant. Why haven't I thought of finding the acceleration of man with respect to the ground before? If a =...

Homework Statement There is a balloon of mass Mb. A rope ladder of negligible mass is hung from it. A man of mass m stands on the rope ladder. A buoyant force F acts on the balloon, causing the man-balloon-ladder system to accelerate upwards. Now, the man climbs up the rope ladder towards...