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-First line is the result in the previous part, with a=2,b=3. -Then I fix n=79 to get a large power of 2 in my expression. -Then I multiply by 2 to get the correct power of 2 in my expression. -Then I subtract 9*105 from my expression. This preserves divisibility by 9.

9|2^{2n}+6n-1 Set n=79, so 9|2^{158}+6*79-1 => 9|2^{159}+6*158-2 => 9|2^{159}+946 => 9|2^{159}+1+9*105 => 9|2^{159}+1.

A tutor so fixated on their way of doing things that alternate approaches/partial solutions are considered wrong and the student is steered towards the tutors method. Even if a students solution/proof is incomplete, the tutor absolutely owes it to the student to give an answer as to whether or...

Re: Extracurricular Integration Marathon \frac{1}{n(n+1)(2n+1)}=\frac{1}{n}+\frac{1}{n+1}-\frac{4}{2n+1}\\ \\ \Rightarrow S_1(N)=h_N+h_{N+1}+3-4\sum_{n=0}^N\frac{1}{2n+1}\\ \\=3-4(h_{2N}-h_N)+o(1)\rightarrow 3-4\log(2) $by identification as a Riemann sum.$\\ \\ S_2(N)=-3+4\sum_{n=0}^N...

Re: MATH2601 Linear Algebra/Group Theory Questions In an inner product space, the norm is defined in terms of the inner product. This means that any operator that preserves the inner product will preserve the norm. Less obvious is the fact that in an inner product space, the inner product can...

Re: Extracurricular Integration Marathon This is pretty good. From memory when I last computed this it was via a similar complex analytic factorisation method, and I believe there was also a hackier solution via ODES. Will look for a nicer way to do it when I have a bit more free time, I...

Re: Extracurricular Integration Marathon...

Re: MATH2601 Linear Algebra/Group Theory Questions Why do you think that "uniqueness result" is true? Add anything orthogonal to x to y and you won't change the inner product of x with it. (Also the truth of b) is (perhaps surprisingly) dependent on the field your vector space is over.)

Re: Extracurricular Integration Marathon We should add infinite series to this thread! \sum_{n=0}^\infty\frac{1}{1+n^2} As usual, "lower tech" methods are preferable.

Register for what? Typically for mathematics, you get invited to selection/training camps based on performance in feeder competitions like the AMC, UNSW etc. There are also the extension programs that used to be run by Geoff Ball (rip :( ). Idk if physics works differently. You will likely...

There is no universally right answer to this, it entirely depends on how on top of your regular courses you are. If you are disciplined and a good student you will probably be fine, but I cannot say that this is true for everyone. Are you currently in the mix for the olympiad team selections...

Re: Several Variable Calculus (The notation Nabla dot (f,g,h) is the divergence of a vector field and is understood in the same way.)

Re: Several Variable Calculus Nabla phi I am sure you understand, that is the gradient of phi and is a vector-valued function. Which can be regarded as a triple of functions (f,g,h) from R^3 to R. Nabla cross (f,g,h) denotes the curl of the vector-valued function (f,g,h). If you write Nabla...

For a set of what? It is basically encoded in the definition of the real numbers that any bounded set of real numbers will have a supremum/infinum. The precise proof of this fact would depend on how you rigorously defined the reals. (And the nonrigorous high school treatment of the reals does...

Re: International Baccalaureate Maths Marathon the question asked you to find where the function's gradient is decreasing, not where the function itself is decreasing.

Re: Extracurricular Integration Marathon Can you post such a formula if you locate it? This particular example posed by omegadot was much nicer than the general case (although as I remarked, the complex method I had in mind generalises well). The indices worked out perfectly modulo 3 to make...

Re: Extracurricular Integration Marathon Kinda funny how often the Basel sum crops up in these things. It's pretty fortunate, because it is not a trivial sum to evaluate, it just happens to be burned into all of our brains more than any series of comparable difficulty due to its fame.

Re: Extracurricular Integration Marathon $Actually, this works:\\ \\ $\int_0^\infty \frac{\log{x}}{x^3+1}\, dx\\ \\ = \int_0^1 \frac{1-x}{x^3+1}\log{x}\, dx.\\ \\ $(Split into integrals over $(0,1]$ and $[1,\infty)$, and substitute $y=x^{-1}$.)\\ \\$\Rightarrow I=\int_0^1 \frac{\log{x}}{x+1}\...

Re: Extracurricular Integration Marathon Yeah, if I look at it later today (I probably will at some point, seems more fun that thesis editing / writing assignment for my course lol), differentiation under the integral / rewriting as a double integral in a clever way is absolutely the main way...

Re: Extracurricular Integration Marathon Couldn't spot a real method, but when computing this in a rather unimaginative complex way I was surprised to discover that my method yielded values for \int_0^\infty f(x)\log(x)\, dx for pretty arbitrary rational functions f(x) (of course with...