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Showing posts from October, 2018

### Twitter proof: the sum of inverses diverges

PtEn< change language In this post I will share with you my favourite proof that the series of the inverses diverges: $\sum_{i=1}^\infty \frac1i = \infty$.

Claim : the series $\sum_i \frac1i$ diverges.

Twitter proof: consider the series \begin{align} &\frac12 + \frac12 + \frac12 + \cdots = \\ &\frac12 + 2 \times\frac14 + 4\times \frac18 + \cdots = \\ &\frac12 + \frac14 + \frac14 + \frac18 + \cdots \leq \\ &\frac12 + \frac13 + \frac14 + \frac15 + \cdots \end{align} that clearly diverges because it is a series of a constant nonzero term. By the comparison test, the series of the inverses also diverges.

Comment with your favourite way to prove this fact!! Neste post quero partilhar com todos a minha prova preferida de que a série dos inversos dos naturais diverge: $\sum_{i=1}^\infty \frac1i = \infty$.

Proposição: a série $\sum_i \frac1i$ diverge.