Let $A \in M_n(\Bbb R)$. How can I prove, that 1) if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2) if $A$ is orthogonal, the...

The standard logarithm expansion is given by \begin{equation} \log(1-x) = -\sum_{n=1}^{\infty}\frac{x^n}{n}, \qquad |x|<1 \end{equation} I have a sum w...

I am redoing exams as a preparation and I found this weird particular exercise to me. "Does $32$ have a multiplicative inverse in modulo $77$? If yes, cal...

Suppose that $x,y,z$ are positive integers satisfying $x \le y \le z$, and such that the product of all three numbers is twice their sum. What is the sum ...

I recently pulled out my old Real Analysis textbook and noticed something that didn't stand out when I was taking the class all those years ago. When...

I would like to show that the Circle group $SO(2)$ is a topological group under multiplication. The requirements are $\alpha$) The group operation is cont...

How to show that the series $$ \sum_{n=1}^\infty (\sqrt[n]{2}-1)$$ diverges ?

Evaluate the roots of $$x^3-6x-6=0$$ I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions a...

I have a function $f \colon R^3 \to R$. I want to find $x$, $y$, $z$ such that $f(x,y,z)=0$. I'm using the method from here: http://en.wikipedia.org/...

So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ But why does it equal this? I learned this in maths not 2 weeks ago and...