Twitter proof: the sum of inverses diverges


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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.

Prova num tweet: considere-se a série$$ \begin{align} &\frac12 + \frac12 + \frac12 + \cdots = \\ &\frac12 + 2 \times\frac14 + 4\times \frac18 + \cdots = \\ &\frac12 + \frac14 + \frac14 + \frac18 + \cdots \leq \\ &\frac12 + \frac13 + \frac14 + \frac15 + \cdots \end{align}$$ que diverge claramente porque é a série em que se soma um termo constante diferente de zero. Pelo teste da comparação, a série dos inversos também diverge.

Comentem qual a vossa forma preferida de provar este facto!!

  - RGS

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