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Alternatively, we can also consider that the general solution of ##F_{n+2} = F_{n+1} + F_n## is $$\begin{split} F_n &= A \left( \frac{1...

A slight correction is indeed required. From ##a_n \geq 1## we have ##a_n \varphi \geq \varphi ## so that $$ |a_{n+1} - \varphi| \leq...

There is a much easier way to arrive at these results. From ##a_{n+1} = 1 + a_n^{-1}## and the definition of ##\varphi## we can obtain...

So that we understand the proofs, and can satisfy ourselves that they are indeed correct proofs. We don't take other people's word for...

It is a fact of basic group theory that conjugation preserves the order of an element (because conjugation is an isomorphism from a...

A generalization is to fix ##g: [0,1] \to \mathbb{R}## and let $$f_n : [0,1] \to \mathbb{R} : x \mapsto \begin{cases} ng(nx) & x < \frac...

I was going to append this to @chwala's thread here, but thought it deserved a new thread. For ##n \geq 1##, define $$f_n : [0,1] \to...

Consider also ##e^{-n\sin \theta}## for ##\theta \in [0, \pi]##.

Responses, like the original post, should be typed and make use of LaTeX rather than screenshots.

I was working at the UK's Ministry of Defence just over 10 years ago when they rolled out Office 365. This involved the MOD making...

One can always resort to the Newton-Raphson method to solve ##\mathbf{f}(\mathbf{x}) = \nabla L = 0## by iteration, where each step...

If $$L = (y_1(x_1) - y_2(x_2))^2 + (x_1 - x_2)^2$$ then the partial derivatives are $$\begin{split} \frac{\partial L}{\partial x_1} &=...