Small cases lie — page 4
Analysis
An error with an unknown in it
Taylor's theorem does not say a partial sum is close to anything. It says the error is one more derivative evaluated somewhere nobody can name, and everything the theorem is worth comes from what happens when that somewhere is replaced by the worst case.
Analysis
The points that ruin the fit
A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.
Applied
Two equilibria and no way to choose
A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.
Dynamics
A carpet with two dimensions
For every set on this ladder so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.
Probability
Giving up on the best
The secretary rule treats landing the second-best exactly as badly as landing the worst, which is a strange thing to want. Ask instead for the smallest average rank and the answer is about the fourth-best candidate — whatever the size of the field, and whether it is ten or ten million.