## Convergence of Series

If you’re taking Calculus II and just learning about sums of sequences — aka series — here’s how I heuristically guess at a problem before I break it down:
$\dpi{300} \bg_white \begin{matrix} \sum_{\infty} {1 \over n} = \infty \\ \\ \sum_{\infty} {1 \over n^{1.1} } < \infty \\ \\ \sum_{\infty} {1 \over n \, \log n} = \infty \end{matrix}$

The last one is the most surprising.  Just remember: log is really, really … really, slow!

It also never stops — look at log x / x, i.e. log versus straight-line, i.e. log per unit.

Of course, you already knew that!  Because

$\dpi{300} \bg_white \mathrm{D} \, [\log x] = \mathrm{flip \ } x = {1 \over x}$.

So just like {1/3, 1/4, …, 1/66, …, 1/7293, …} never settles down to zero, thus log never stops increasing.  But all the while, log is increasing ever more slowly.

NOTE TO PEDANTS: You might object that ∞ is “not a number” so I can’t use the equals sign.  To you I say,
(a) consider using hyperreal or surreal numbers;
(b) consider projective geometry;
(c) consider the Riemann sphere.

All three use the point ∞ as an element of the set of numbers.

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