savoy082

Pt En In this twitter proof we will see that no polynomial grows faster than the exponential function. Claim: the ratio $\frac{x^p}{e^x}$ tends to $0$ as $x $ tends to infinity. Twitter proof: recall that by the Taylor expansion of $e^x $ we have $$e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$$ and for $x > 0$ we have $$\frac{x^p}{e^x} \leq \frac{x^p}{\sum_{i=0}^{p+1} a_ix^i} \to 0 $$ where $a_{p+1} \neq 0$ thus proving our claim. The way I like to look at this is "if the exponential has a bit of every polynomial inside, then it will grow faster than any fixed polynomial $p(x)$" (because, in particular, the exponential "has a bit" of all other polynomials that have degree higher than that of $p(x) $. Neste post vamos ver que a função exponencial cresce mais depressa que qualquer função polinomial. Proposição: o rácio $\frac{x^p}{e^x}$ tende para $0$ quando $x $ tende para infinito. Prova num tweet: sabemos pela expansão de Taylor de $e^x $ que $$e^x = \sum...

Pt En In this twitter proof we will see how the exponential function can mess up with objects from our daily lives!.. Claim: with less than $50$ folds, a piece of paper will be so thick that it will cover the distance from the Earth to the Moon. Twitter proof: a common sheet of paper is $0.1$mm thick. If we fold it once, it becomes $0.2$mm thick. Folding twice, $0.4$mm. Folding $49$ times, the paper becomes $2^{49}\times 0.1$mm thick, which is around $5.63\times 10^{13} $mm or $5.63\times10^7$km, $141$ times the distance from the Earth to the Moon ($398818$km). Neste post vamos ver como a função exponencial pode interagir com objetos do nosso quotidiano e criar resultados inesperados. Proposição: com menos de $50$ dobras, uma folha de papel fica com uma grossura superior à distância da Terra à Lua. Prova num tweet: uma folha de papel normal tem $0.1$mm de grossura. Se a dobrarmos uma vez, fica com $0.2$mm de grossura. Dobrando de novo, fica com $0.4$mm. Dobrando $49$ vezes,...