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:
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
.
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.
