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Re: Discrete Maths Sem 2 2016 The question can be rephrased as asking: "What is the probability of a randomly chosen f:X->Y being surjective if |X|=k and |Y|=n?" The total number of functions between these sets is n^k. The number of surjective functions arises from partitioning the |X| into n...

If you are approaching these kinds of facts as identities to memorise, you are going to make life hard for yourself as there are too many to count! Instead, when you learn about a new object/operation, make sure that the definition is really ingrained in your head (multiple equivalent...

What's with the dx's? lol. And are the a_k real numbers? If so, the answer is no (consider the alternating harmonic series). If the a_k are positive real numbers, the answer is yes.

Re: HSC 2016 4U Marathon - Advanced Level He isn't "restricting the means" of resolving the question, he is just saying that a solution that jumps in halfway through after an unjustified reduction of the problem is not a complete solution. (And I agree with him.) I don't think the leap from...

Re: MATH1231/1241/1251 SOS Thread s_n^2 \leq \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n \frac{1}{k^2}\right). Each of these factors is a partial sum of a convergent positive series and so is bounded above by its limit. Hence s_n is bounded.

Re: MATH1231/1241/1251 SOS Thread If applying the linear operator A to the vector v scales it by lambda each time, and we apply A to v k times...

There is an absolute value in the change of variables for the Jacobian because of the subtle issue of orientation. When we write something like \int_S f \, dx for S a subset of R^n and f: S -> R, this is an unoriented notion of integration. (Because S is a subset, without choice of...

Huh? Can you elaborate on what you are trying to do by splitting the sum in this way? This is not a straightforward application of the p-test, because of the presence of the factorial and also because the exponent is not fixed. It is a basic application of the ratio test though. (As questions...

On this topic, a good exercise for your intuition is trying to prove that the ratio test is strictly weaker than the root test. I.e. let x_n be a given positive sequence. a) Show that if the ratio test asserts that the sum of x_n is convergent, then the root test does too. b) Show (by way of...

Check your calculations, the ratio test is conclusive. (The ratio tends to exp(-1).)

I won't do the calculation for you but here is the idea: The -1-eigenspace and the 5-eigenspace are orthogonal. (This can be seen directly by computing dot products or as a consequence of A being symmetric.) So being given these three linearly independent eigenvectors v1,v2,v3, we need to find...

Re: MATH1231/1241/1251 SOS Thread Why should the top be negative in the large n limit? The trig factor oscillates and changes sign. This fraction is boundedly divergent because the size of the numerator and denominator are both ~n where "~" formally means "bounded between constant positive...

Linear Algebra Marathon & Questions This is a marathon thread for linear algebra. Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute. (mod edit 7/6/17 by dan964)...

$Am assuming these sequences are positive so we don't need to take roots of negative numbers. Then I think the following method works.\\ \\ Set $b_n=\max(a_n,2^{-n}).$ Since $a_n\leq b_n \leq a_n+2^{-n}$, it is clear that $\sum a_n$ converges iff $\sum b_n$ converges.\\ \\ So $\sum...

In a sense that double integral (technically that is an iterated integral, but Tonelli's/Fubini's tells us this is the same thing as the corresponding double integral) is how we define area, at least until one encounters measure theory. Notice that this definition of area coincides with the...

It is far too early to know exactly what you will like, so don't base your choice of uni too much on this.

Have had many years of experience at both usyd (did my BSc Adv Math and honours there) and ANU (have been doing my Phd in pure math here for the last few years, and have taught and observed various undergrad courses). They are both strong programs and turn out many fine graduates to both...

Lagrange multipliers is how you deal with constrained optimisation problems. I.e. when you are optimising a function f on a hypersurface in R^n rather than on R^n itself. In this case however, we were optimising a function over a domain in R^n. You can optimise the function in the interior of...

Re: First Year Uni Calculus Marathon Sure. $For $x\in [0,1]$ $\\ \\H(x+n)-H(n) \leq H(n+1)-H(n)\\ \\ \Rightarrow \lim_{n\rightarrow\infty}\int_0^1 H(x+n)-H(n)\, dx = 0\\ \\ \Rightarrow \lim_{n\rightarrow\infty}\left(\int_0^1 H(x)+\sum_{k=1}^n \frac{1}{x+k}\, dx - H(n)\right)=0\\ \\...

Re: First Year Uni Calculus Marathon \int_0^1 H(x)\, dx\\ \\ =\int_0^1\left(\int_0^1 (1-t^x)\sum_{k\geq 0}t^k\, dt \right)\, dx\\ \\ =\lim_{n\rightarrow\infty}\int_0^1\left( \int_0^1 (1-t^x)\sum_{k= 0}^n t^k \, dt\right) \, dx\quad (\star)\\ \\ =...