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Re: First Year Uni Calculus Marathon Bump. Fun fact, am currently proving a predictable generalisation of this theorem's big brother for use in attacking a thesis problem. Will give a brief outline to show how easier problems can lead to pretty strong and useful results. So...

Re: First Year Uni Calculus Marathon That is what makes this problem interesting though, the (a_n) are unspecified. If the (a_n) were (say) bounded, then the Taylor series would have a positive radius of convergence, and from some basic analysis, the limiting sum would be a smooth function...

Re: First Year Uni Calculus Marathon The coefficients of x^k in what? It is true that: f_n(x):=\sum_{k\leq n}\frac{a_kx^k}{k!} has f_n^{(j)}(x)=a_j for 0\leq j \leq n, but the formal expression \sum_{k\in \mathbb{N}}\frac{a_kx^k}{k!} need not converge anywhere apart from x=0...

Re: First Year Uni Calculus Marathon Taylor series are indeed the correct lines to think along here, but as an arbitrary Taylor series need not converge anywhere apart from the point it is centred at, this problem is not completely trivial. Do you have a construction in mind?

Re: First Year Uni Calculus Marathon Here's a cool one to think about: $Given an arbitrary sequence $(a_k)_{k=0}^\infty$ of real numbers, does there exist a a smooth function $f\in\mathcal{C}^\infty(\mathbb{R})$ such that\\ \\ $\frac{d^kf}{dx^k}(0)=a_k$ for each $k$?$\\ \\

Re: Discrete Maths Sem 2 2016 There are lots of ways. Firstly note that you can assume that m and n are non-negative without loss of generality. You can then do it using the fact that (m-n)(m+n)=2. An alternative approach is based on something sometimes called the discrete inequality. If...

Re: First Year Uni Calculus Marathon J(t):=\int_0^\pi \frac{\log(1+t\cos(x))}{\cos(x)}\, dx\\ \\ = J(0)+\int_0^t J'(s)\, ds\\ \\ = \int_0^t \left(\int_0^\pi \frac{d\phi}{1+s\cos(\phi)}\right)\, ds\\ \\ = \int_0^t \frac{\pi}{\sqrt{1-s^2}}\, ds\\ \\ = \pi \sin^{-1}(t) $ for $t\in [0,1].\\ \\...

Re: Discrete Maths Sem 2 2016 That doesn't make any sense, when you talk about a function, you have to specify a domain and a codomain for that function. Loosely speaking a function is a pair of sets X,Y and a rule that assigns to every element of X an element of Y. The range is the set of...

Re: Discrete Maths Sem 2 2016 "Codomain was a proper subset of f"? What do you mean by this? Yes to your second question. Read that kind of arrow as "maps to".

Re: Australian Maths Competition 2013 Uh, if a,b,c,m are positive integers, how can we possibly have am^2+bm+c=f(m)=0?

Yep, that's the danger of colloquial statements like: "The sum of the positive integers is -1/12". They are catchy titles for articles/videos/blogposts but are very imprecise and misleading.

To be able to say that the statement is "true" we need to come up with a way of making sense of a divergent series like this one. This is the real issue of the question, because there are multiple ways of doing this. The most obvious such method makes use of the analytic continuation of the...

Re: Discrete Maths Sem 2 2016 Yep, it really isn't. As it happens (A*B)*C=A*(B*C)=The set of things that are elements of exactly one of the three sets.

Re: Discrete Maths Sem 2 2016 Have you learned that the symmetric difference is associative? If not, then prove this as an exercise as it is easy and has this claim is a trivial consequence. Indeed, We have A*C=B*C => (A*C)*C=(B*C)*C => A*(C*C)=B*(C*C) => A*e=B*e => A=B where e denotes the...

Re: Discrete Maths Sem 2 2016 Another way of doing it that doesn't involve taking elements is: A=(A\cap C)\cup (A\cap C^c)\subseteq (B\cap C)\cup ((B\cup C)\cap C^c)= (B\cap C) \cup (B\cap C^c) \cup C\cap C^c =B. Where the \subseteq step follows from applying the question assumptions to...

Re: HSC 2016 4U Marathon - Advanced Level And a more algebraic question: $For a function $f:\mathbb{N}\rightarrow \mathbb{N}$, we define it's \emph{forward difference} by $(\Delta f)(n)=f(n+1)-f(n)$. (In many ways, $\Delta$ is a discrete analogue to differentiation, where $(Sf)(n):=\sum_{k\leq...

Re: Discrete Maths Sem 2 2016 Don't view "," and "|" in the same way (btw ":" is a common alternative for "|" that is my personal preference). The former is basically informal formatting here, whilst the latter is part of the formal set builder notation syntax. {x in A: Mathematical statement...

Re: First Year Uni Calculus Marathon Are you talking about subsets of the reals/complex numbers with the standard operation of addition? Then the only finite additively closed set is {0} because if some nonzero z is in it, then nz:=z+...+z (n copies) must be in it for all positive integers n...

Re: HSC 2016 4U Marathon - Advanced Level Written as intended, in step 2 you are only dealing with finite sums and integration is not the only thing you are doing in step 4. I just didn't want to be too explicit/prescriptive about the limiting argument, as methods will vary depending on level...

Re: HSC 2016 4U Marathon - Advanced Level Here is a derivation of \zeta(2) via a scenic route. $1. Prove that: $\int_0^\infty \frac{\sin(x)}{x}\, dx=\pi/2$ by considering the related integral $ \int_0^{\pi/2}\frac{\sin((2n+1)x)}{\sin(x)}\, dx $2. Find an integral expression for...