$ \color{goldenrod}{\text{Th}} $: Consider sequence $ (a_j) $ in $ \mathbb{R}_ {\geq 0}$. Now $ \displaystyle \sum_{j=1}^{\infty} a_j $ converges if and only if $ \displaystyle \sum_{j=1}^{\infty} \dfrac{a_j}{1+a_j} $ converges. $ \color{goldenrod}{\text{Pf}}$: $ \underline{\implies} $ Clear as $ \frac{a_j}{1+a_j} \leq a_j $. $ \underline{\im... Read more 06 Aug 2021 - 1 minute read
$ \color{goldenrod}{\text{Th}} $: Consider series $ \sum_{n=1}^{\infty} v_n $, where $ (v_n) $ is a seq in a complete normed space. Let $ \ell $ $ := \limsup_{n\to\infty} |v_n|^{\frac{1}{n}} \in [0, \infty] $. Now : If $ 1 < \ell \leq \infty $, the series diverges. If $ \ell < 1 $, the series converges absolutely. $ \color{goldenrod}{\... Read more 06 Aug 2021 - 5 minute read
$ \color{goldenrod}{\text{Th}} $: For any $ \sigma \in S_n $, we have $ \text{ord}(\sigma) \leq (e^{\frac{1}{e}})^n \text{ } ( \leq 1.445^n ). $ $ \color{goldenrod}{\text{Pf}} $: Let $ \sigma = \sigma_1 \ldots \sigma_k $ be decomposition into disjoint cycles, and $ | \sigma_i | =: \ell_i $. Now $ \text{ord}(\sigma) $ $ = \text{LCM}(\ell_1, \ldo... Read more 06 Aug 2021 - 1 minute read
$ \color{goldenrod}{\text{Th}}$: Take a continuous map $ [a,b] \overset{f}{\to} \mathbb{R} $, such that $ f’’ $ exists and is $ > 0 $ on $ (a,b) $. Then $ f((1-t)a+tb) < (1-t)f(a) + tf(b) $ for all $ t \in (0,1) $. $ \color{goldenrod}{\text{Pf}}$: LHS is just $ f(x) $ evaluated at $ (1-t)a + tb $. RHS is just “line” $ f(a) + \left( \dfr... Read more 06 Aug 2021 - 1 minute read
Consider $ f : \mathbb{R} \to \mathbb{R} $ with $ f(x) = 0 $ at irrational $ x $ and $ f(\frac{p}{q}) = \frac{1}{q} $ at rationals. This is Thomae’s function. Btw when we say “a rational $ \frac{p}{q} $” it will be implicit that $ p, q $ are integers, $ q > 0 $ and the fraction is in reduced form. $ f $ is discontinuous at rationals. ... Read more 05 Aug 2021 - 1 minute read