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[ZetaX]

You can use Stirling's formula to get a good approximation of x=n! by sqrt(2 pi n) (n/e)^n. Then you need to solve for n. You can do this by Newton's algorithm or any other way to find zeros of analytic functions. Taking the logarithm first and observing that ln(x)=ln(n!) is about n ln(n) - n might simplify it a bit, at the cost of accuracy.
After that, you just need to find the exact n where you get bigger than x by searching around the solution you got.

Generally, if your function is monotonous, then you only need to find integers M,N with M < x < N and then use binary search in [M,N].

[Pnx]

I see I've managed to ask one of those questions that's much harder to answer than I thought it was. 

[Pnx]

Ok, so I've got the sum of (2/3*5) + (2/7*7) + (2/11*9) . . . Which I wrote as 2/(5k+3)(2k+5) for k=0 to infinity.

I'm pretty sure it converges on 1/4 or something like that, but I'm having trouble getting that answer. Can someone help?

EDIT: Oh, and before I got bed I've got the sum of k^1/2ln(2k) / (k³-4). Don't need the sum, just need to know convergence/divergence. Pretty sure it converges but I can't find a test that works.

[da_nang]

Ok, so I've got the sum of (2/3*5) + (2/7*7) + (2/11*9) . . . Which I wrote as 2/(5k+3)(2k+5) for k=0 to infinity.

I'm pretty sure it converges on 1/4 or something like that, but I'm having trouble getting that answer. Can someone help?

EDIT: Oh, and before I got bed I've got the sum of k^1/2ln(2k) / (k³-4). Don't need the sum, just need to know convergence/divergence. Pretty sure it converges but I can't find a test that works.

Well, for the first one I got (16-3*π)/42 by rewriting the fraction 2/((4k+3)(2k+5)) as partial fractions and then transforming them to integrals of sums of polynomials. However, WolframAlpha claims I'm missing an added ln(64)/42 in there and I can't for the life of me find where I'd missed it.

For the second one, note that k^1/2ln(2k) / (k³-4) = k^1/2ln(k) / (k³-4) + k^1/2ln(2) / (k³-4)

k^1/2 / (k³-4) <  k^1/2 ln(k) / (k³-4)  < (k - 4^1/3) / (k³-4) = 1/(k² + 4^1/3k + 4^2/3) which converges by the integral test.

Thus k^1/2ln(2k) / (k³-4) converges as well by comparison.

[Pnx]

For the first one I'm not actually sure how you set it up as a sum of polynomials. Which is sort of my problem. I believe I need some method of creating a function for the nth term but I haven't been able to.

For the second... You don't seem to have actually established that (k - 4^1/3) / (k³-4) > k^1/2ln(2k) / (k³-4).

[ZetaX]

For any fixed eps you have that ln(n) < k^eps for all k large enough. Use this with eps=1/2 and you get majorisation by some multiple of 1/kČ.

[Pnx]

I really don't understand what you just wrote.

[da_nang]

For the first one I'm not actually sure how you set it up as a sum of polynomials. Which is sort of my problem. I believe I need some method of creating a function for the nth term but I haven't been able to.

For the second... You don't seem to have actually established that (k - 4^1/3) / (k³-4) > k^1/2ln(2k) / (k³-4).

1. 1/P(k) = integral from 0 to 1 of x^{P(k) - 1}, where P(k) is a positive polynomial of degree n > 0. Moving the summation inside the integral results in an integral of a series of polynomials.

For instance, when P(k) = 4k+3, summing the x^P(k)-1 terms evaluates this infinite sum to x²/(1-x⁴) and for P(k) = 2k+5, the sum evaluates to x⁴/(1-x²). The difference of these rational functions are integrated (as was determined by the partial fraction decomposition). Note I left out the coefficients of the polynomials in the integrands for clarity.

2. Any polynomial of degree n > 0 grows faster than the logarithm. In Big-O notation, O(ln(x)) < O(x^k), k>0

[ZetaX]

I really don't understand what you just wrote.

Then you should go back to your material and reread it, as those type of statement ("epsilontics") is very standard and essential to understanding (actually: defining) limits. The property about ln(x) can easily be proven, you should try yourself (after checking the epsilon stuff).

[Lightningfalcon]

Not exactly math, but it has math in it and I can't find a chemistry thread.  I don't care about the actual answers, all I want to know is how to do them.  Preferably by tommorow morning at 8:15.
How do you determine the order of a reaction by looking at the graph?
If the decomposition of N₂O₅ to NO₂ and O₂ is first order with a rate constant of 4.8*10-4:
a. If the initial concentration is 1.65*10-2 mol/L, what is the concentration after 825s?
b. How long will it take for the concentration to decrease to 1*10-2 mol/L from the initial concentration given in a?
d. What is the half life of the reaction?


This is the only part of the review sheet I can't get, which is mostly my fault due to lack of sleep. (Screw essays of any and all kind). 

[Skyrunner]

We had a huge debate about whether the Coriolis effect is inertial force o.O

[MagmaMcFry]

We had a huge debate about whether the Coriolis effect is inertial force o.O

An inertial force is a force that appears in non-inertial reference frames as a correcting factor for Newton's laws, so the Coriolis effect is in fact an inertial force. What's there to argue about?

[Skyrunner]

It's because the textbook definition of inertial frame of reference, inertial force, and inertia is confusing. Also, the words for "frame of reference" and "system" are identical.

Heck, in the end the teacher concluded that the Coriolis effect isn't inertial!

[MonkeyHead]

Hey, don't blame Physics, blame language for being shit at representing Physics.

[Furtuka]

​Does anything worth writing up happen when you start with a 3D vector field Field[x, y, z] and calculate the curl of curlField[x, y, z] which is given by
curlcurlField[x, y, z] = ∇⨯(∇⨯Field[x, y, z])?​