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$First, we observe that since: $\frac{2Y_n}{2X_n+1}\leq \frac{Y_{n+1}}{X_{n+1}}\leq \frac{2Y_n+1}{2X_n}$, and both sequences tend to $\infty$, it suffices to establish the result for one parity.\\ \\ To this end, we define: $\bar{X}_n=X_{2n-1},\bar{Y}_n=Y_{2n-1}.$\\ \\ The recurrences in the...

I think 5/2, will write proof in a sec. $First, we observe that since: $\frac{2Y_n}{2X_n+1}\leq \frac{Y_{n+1}}{X_{n+1}}\leq \frac{2Y_n+1}{2X_n}$, and both sequences tend to $\infty$, it suffices to establish the result for one parity.\\ \\ To this end, we define...

That's the thing though, you shouldn't just be absorbing the solution techniques you read in books. When you come across hard problems you should bang your head against them for a bit and try to come up with clever ways of doing them yourself. Success or failure, this way you get a MUCH better...

Don't choose a tutoring place based on how many state ranking students went there. (And don't choose to get tutoring just because some state ranking students do.) Most SR students would get SRs or close to, regardless of what tutoring they received and indeed whether or not they received...

Re: First Year Uni Calculus Marathon And here's a new one simpler than all three of the above. $Suppose there exist positive constants $C,\rho$ such that $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the inequality $|f(x)-f(y)|\leq C|x-y|^{\rho}$ for all $x,y\in \mathbb{R}$.\\ \\ a) For which...

Re: First Year Uni Calculus Marathon Bumping some unanswered questions of mine. The third is a bit harder than the first two.

Re: HSC 2016 MX2 Combinatorics Marathon As we have had no takers yet, I will break this down. A strategy in this problem is a method for decisionmaking that can tell you whether or not to accept the k-th candidate, given only the information of how this candidate stands in relation to the...

Re: HSC 2016 MX2 Combinatorics Marathon Well, you might as well assume that it depends on n (a priori, why shouldn't it?) and find it in terms of n then. I don't want to spoil this problem for anyone as it is a nice one.

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread Why are you surprised that the midpoint of two points is equidistant from these two points?

It's not really advanced enough to be classified as either imo. It is a collection of tools most of which are useful to people doing either pure or applied work in uni / their career. Some of these tools are motivated by real world applications (super straightforward differential equations in...

I think it is just 1. (Achieved for example, when a=b=c=1.) P=\sum_{cyc}\frac{a}{a+\sqrt{3a+bc}}\\ \\ =\sum_{cyc}\frac{a(a-\sqrt{3a+bc})}{a^2-(a+b+c)a-bc}\\ \\ =\sum_{cyc}\frac{a\sqrt{(a+b)(a+c)}-a^2}{ab+ac+bc}\\ \\ \leq \sum_{cyc} \frac{a(2a+b+c)/2-a^2}{ab+ac+bc}\\ \\=1.

Re: First Year Uni Calculus Marathon s is a variable used only in the definition of the Gamma function, n refers to dimension.

Re: First Year Uni Calculus Marathon The one written in the question.

Re: First Year Uni Calculus Marathon Here is an exercise to exhibit a couple of quirks of high dimensional solids. (Not as directly related to first year calculus courses as most things in this thread, but it is first year level stuff, and it is in the calculus/analysis ballpark.) a) By...

Re: Extracurricular Integration Marathon Just to demonstrate that it is indeed an alternating version of the Basel sum, to people who would not recognise it. (I certainly don't know/remember that many of these identities off the top of my head myself.)

Re: Extracurricular Integration Marathon I know you asked to avoid the use of power series, so I will figure out another method for the simpler integral where I use it. In the meantime though: $Set $F(t)=\int_0^1 \frac{\log(1+tx^2)}{1+x}\, dx\\ \\ \Rightarrow F(1)=\int_0^1 F'(t)\, dt\\ \\...

Re: Extracurricular Integration Marathon Note: The above idea is similar in spirit to how we can bound sin/cos between their partial Taylor series of opposite parity.

Re: Extracurricular Integration Marathon Have been a bit busy the last couple of days, hence the delay. The method does indeed work and is based on the estimate: \left|\int_{TM}^\infty \phi(x)p(x)\,dx\right| \leq p(TM)\left|\int_{TM}^{T(M+1)}\phi(x)\, dx\right| if p monotonically decreases...

Re: First Year Uni Calculus Marathon S_n=\sum_{k=0}^n (1-\frac{k}{n})^n=\sum_{k=0}^\infty u_{n,k} where u_{n,k}=0 for k>n, and u_{n,k}=(1-\frac{k}{n})^n otherwise. As n\rightarrow \infty, u_{n,k}\nearrow e^{-k}, the monotonicity of this convergence following from basic calculus / other...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread On the topic of binomial stuff $Prove that: $\sum_{k=0}^n \binom{m+k}{m}=\binom{m+n+1}{m+1}.