I actually misread and misunderstood your post chinagambler. So this is incorrect. kamikaze baby's post is correct below. This got me curious and I played around with Excel for a bit. There's a way to do it in Excel which I put below.

There actually is a formula for Excel to compute it. You want to use BINOM.DIST(number_s,trials,probability_s,cumulativ e)

The first slot is number of successes, the 2nd slot is the number of trials, the 3rd slot is the probability. The fourth slot is TRUE/FALSE. If it is FALSE then the function spits out the probability that there are exactly number_s successes. If it is TRUE then it returns the probability that there are AT MOST number_s successes.

So if you want the chance that an event with a probability p of occuring will occur at least x times out of n trials I think the formula you want in Excel is =1-BINOM.DIST(x,n,p,TRUE)

Hey, this isn't a homework question but it was a question on my last midterm and I kept getting stuck on proving whether this converges or diverges. I tried several different tests but it seems like I was going in circles. Just was interested in seeing the answer, thanks.


hosting images

http://math.stackexchange.com/questi...n-dense-in-1-1

This should help do the trick.

HUStylez, that seems like a devilish problem to put on a midterm. I've thought a bit about it and think it diverges but am having trouble coming up with a rigorous solution (perhaps there's some convergence rule I've forgotten?); maybe writing sin(n) as (e^{in} - e^{-in})/2 and expanding will work.

Wyman, perhaps I'm being dense but how does what you linked to help? Does it somehow imply that the terms (sin n)^n are not tending to zero?

http://mathoverflow.net/questions/54...eries-converge

Not so easy !

Yes, my plan was to use the density of sin(n) in [-1,1] to bound sin(n)^n away from 0. But I had not worked out the details. Surely there's a cleaner proof than the one slipstream linked to, but I know many of the people in that thread and it's hard to believe they'd have missed one!

I'll keep thinking about a more elementary proof.

Hey,

with Abb(R,R) I mean all functions from R-->R

U=(f element of Abb(R,R) with f(x)=0 for almost all x element of R)

1) Proof that U is a subspace of Abb(R,R)[this one is easy, I already have that]

2) Find a basis for U

We haven't talked about basis of functions yet, so I do not really have a clue here.

I don't think the density alone will get you there. My intuition (which, admittedly, has been wrong many times before) is that there exist series {a_n} dense in [0,1] with {a_n^n} tending to 0, so long as you arrange the elements {a_n} in a clever way.

The density alone should not, I agree, although I have spend a small amount of time trying to construct a counterexample and failing

For example, take the rationals in (0,1), and arrange them such that every term a/b with a > 1 appears at term b^2 or later. Like:
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 2/3,
1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 3/4,
1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 2/5, 3/5, 4/5,....
Let this series be c_n. Then the series {c_n^n} tends to zero. To see this, note that obviously the subsequence of terms of the form 1/b is tending to zero. For the other terms, for c_n = a/b with a > 1 we have
c_n^n < (1-1/b)^{b^2} \to e^{-b}.
Since (n^2)/2 < b (say), this gives
c_n^n < e^{-(n^2)/2}
This seems to imply that the sum also converges.

Thanks for your insight everyone, and for the record it was the "challenge" problem on the midterm, so it was the last question worth the least amount of marks lol, I was just very stumped on it!

The simplest example for a dense sequence s.t. a_n^n is a convergent series is to intercalate a lot of 0s between the terms of a dense sequence (r_k), only putting the r_k's at spots n(k) s.t. r_k^n(k) < say 2^(-k).

Mathematica says it doesn't converge. Wolframalpha says it is

It's a strange problem...I'm pretty sure you can't find a basis for U as a vector space without appealing to the axiom of choice.

Yeah for someone who hasn't talked much about bases this is a strange problem.

If the problem were to find a basis for V, a subset of Abb(R,R) such that each element of V is zero except at a finite number of points in R, then a basis would be {f_a(x): a in R}, where f_a(x) is 1 at x=a and 0 elsewhere.

But this is not a basis for U, since we can form every element of a space by a finite linear combination of basis elements. And elements of U are allowed to be nonzero at infinitely many points in R (though a set of measure zero. Take f(x) = 1 if x is an integer, 0 otherwise, for example).

Find the image of set S under given transformation

S is the disk given by u^2 + v^2 <= 1 x= au y=bv

It seems in similar problems you can come away with exact points for bounds of the image. Here I just get an ellipse depending on what a and b are u = (1,0 becomes x=(a,0) and v= (0,1) becomes y=(0,b) and it traces around from there? Is that correct?

Yeah just substitute: u = x/a, v = y/b

(x/a)^2 + (y/b)^2 <= 1

The equality is the classic form of an ellipse, and this just includes the interior.

Use the given transformation to evaluate the integral



I figured out that it is a rotated ellipse could the jacobian and get v and u in terms of x and y but the equations don't look pretty and I can't figure out the image or really where to go.

Is there any shortcut because the region is in terms of the integrand?

You should actually do the substitution to see what your region is in the u-v domain.

Oh for some reason that algebra looked discouraging but once I chug it all out there is nice cancellation and I get a circle and seems good from there

Evaluate the integral by making appropriate change of variables



The integrand is sin 9x^2 + 4y^2 and the region is an ellipse. Should the sum of a bunch of the sines over a symmetrical ellipse centered at the origin equal zero, or do all of the random values on the inside throw it off somehow? I felt pretty good about doing the problem but I'm trying to check to see if the answers make sense at the end and I don't know if this does ( I got an answer between 0 and 1)

I got (1-cos(1))pi/12

edit: but i did a lot of steps in my head, so if i have a sign wrong or if it should be a 3pi instead of pi/12, that's reasonable.

oh, you were integrating wrt theta from 0 to 2pi right because I just realised I skipped the integration wrt theta and I wind up getting 2pi * (1- cos1) /12

so its not a zero because you are taking the sin of values rather than hitting all of the thetas around the ellipse as you would in a circle. and the squared values might eliminate symmetrical cancelling since its a bunch of positive values

Yeah as soon as you start taking functions of the angles and r, who knows what's gonna happen?

Additionally, it's from 0 to pi/2, right?

oh completely missed that in the first quadrant part so my question made no sense to begin with