Quantcast

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:
\sum_{\infty} {1 \over n}	=	\infty    \\ \sum_{\infty} {1 \over n^{1.1} }	<	\infty   \\ \sum_{\infty} {1 \over n \, \log n}	 =	\infty

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

The Logarithm

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

Log x / x

Of course, you already knew that!  Because

D[ log x ] = 1 / 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.

51 notes

  1. patman023 reblogged this from isomorphismes
  2. isomorphismes posted this