And it has been known since as early as 1350 that this series diverges. Oresme’s proof to it is just so beautiful.

Now replace ever term in the bracket with the lowest term that is present in it. This will give a lower bound on S_{1.}

Clearly the lower bound of S_{1} diverges and therefore S_{1} also diverges.
But it interesting to note that of divergence is incredibly small: 10 billion terms in the series only adds up to around 23.6 !