User:X42bn6/Sandbox

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\forall a_n\in{[\mathbb{R}]}.\left(\begin{array}{l}\left(\displaystyle\lim_{n\to\infty}\left|\displaystyle\frac{a_{n+1}}{a_n}\right|<1\Rightarrow\exists l\in\mathbb{R}.\left(\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=0}^na_i\rightarrow l\right)\right)\\\and\left(\displaystyle\lim_{n\to\infty}\left|\displaystyle\frac{a_{n+1}}{a_n}\right|>1\Rightarrow\nexists l\in\mathbb{R}.\left(\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=0}^na_i\rightarrow l\right)\right)\\\and\left(\displaystyle\lim_{n\to\infty}\left|\displaystyle\frac{a_{n+1}}{a_n}\right|=1\Rightarrow\left(\begin{array}{l}\left(\exists l\in\mathbb{R}.\left(\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=0}^na_i\rightarrow l\right)\right)\\\or\left(\nexists l\in\mathbb{R}.\left(\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=0}^na_i\rightarrow l\right)\right)\end{array}\right)\right)\end{array}\right)