$$\lim_{n \to \infty}(5^n-4^n)$$ My intuition told me that this is infinity, but the proof of this I managed to find is quite ugly $$\lim_{n \to \infty}(5^n-4^n) = \lim_{n \to \infty}4^n((\frac{5}{4})^n-1) = \infty(\infty-1)=\infty$$ I was wondering if there was another (preferable nicer way) of...

Consider the following property of a function $f$: There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function. Question 1: Is there a name in the literature for this property? Question 2: Are there any good references for reading about the class o...

Let $f:\Omega\to\mathbb{R}$ be of $C^k$ class, let $0\in\Omega\subset\mathbb{R}^n$ and let $\Omega$ be star shaped at $0.$ From Hadamard's Lemma we know that we can write function $f$ as $$f(x)=f(0)+\sum_{i=1}^n \overbrace{x_i\int_0^1\frac{\partial f}{\partial x_i}(tx)dt}^{R_i(x)}.$$ This is i...