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I don't think just doing the geometric series will get full marks because there are obviously cases where |b| = 1. A neat way to avoid that though is: |\beta|^n \le M(|\beta|^{n-1} + ... + 1) So: |\beta|^n + |\beta|^{n-1} + ... + 1 \le (M+1)(|\beta|^{n-1}+...+1) but LHS > |\beta|^n +...

the answer to q8biii is wrong, the method this guy uses only counts cases where the repeated ball is drawn a second time on the final draw

wikipedia it, the line Re(z) = -1/2 is very significant lol

once you get |z+1| <= |z| you can do it straight away by considering it geometrically also lol riemann hypothesis much?

This works: \frac{p}{1+p} + \frac{q}{1+q} - \frac{r}{1+r} \\ = (1-\frac{1}{1+p}) + (1-\frac{1}{1+q}) - (1-\frac{1}{1+r}) \\ = 1 - \frac{1}{1+p} - \frac{1}{1+q} + \frac{1}{1+r} \\ \ge 1 - \frac{1}{1+p} - \frac{1}{1+q} + \frac{1}{1+p+q+pq} \\ = 1 - \frac{1}{1+p} - \frac{1}{1+q} +...

Theres a really neat way to do it if you start by observing that x/(1+x) = 1 - 1/(1+x) and 1/(1+r) ≥ 1/(1+p+q+pq) = 1/(1+p)(1+q): the equation then factorises into positive things

8cii was really hard

reaaaaaaalllyyyy unconventional paper

probability is copping so much flak! my favourite questions are circle geo closely followed by perms and combs, least favourite is integration or goddamn curve sketching

oh okay you took out some of the help

did you send me this question in a pm at one point? (its a good question!) but what about its later parts

That questions pretty similar to finding the integral of (sinx)/x isnt it?

crap yeah, doesnt really cahnge the content of the proof though (just replace n with n+1 where necessary)

Okay um do it like this: Let L_n = \sqrt{n^2+n}-n. We're given that this sequence has a limit, also clearly as n < \sqrt{n^2+n} < n+\frac{1}{2}, L_n is always less than half, and it is also clearly always positive. Now rearrange that equation there to be in terms of n: (L_n + n)^2 = n^2+n...

you mean |x| < 1, negative values of x and zero still work to make sure the GP converges, do it like this: S_n = 1+xcis(x)+x^2cis(2x)+...+x^ncis(nx) = \frac{1 - (xcisx)^n}{1-xcisx} So if we let xcisx = z, S_n = \frac{1-z^n}{1-z} which converges as n->infinity if |z| < 1, and diverges if...

Yeah I know what the MVT is, but I don't really get how your proof of seanieg89's question actually works (like how do you assume f'(c) is constant o.O, presumably as you range x and y)

Sorry for digging up an old post to quote but this user did try refute some earlier claims I made, so I may as well start from the claim that speeches are harder than hamlet I'll start by saying I think the speeches are inappropriate for modb in the first place given what modb is supposed to...

I don't really pretend to understand mean value theorem, but I really don't get how your proof is supposed to work math man. Like how do you know that f'(c) or K is a constant? You've chosen a particular c for a particular pair y<x, but wouldnt the desired value of f'(c) vary for the pairs of...

Actually for seanieg89 and mathman, try this question (its of very similar flavour and pretty good as far as I remember it): Let f be a function that takes real numbers to non-negative reals, and define g(x) = \frac{f(x)}{e^x}. Prove that f satisfies the following inequality for all reals x...

I don't know of any, just do what the book says and you'll be fine I assume (I dont have that text book) Do it like, case1: you use two L's and two S's case2: you use two L's and at most one S (here, S is just like any other letter( case3: you use two S's and at most one L case4: at most on L...