Long before XKCD named it, I was a big fan of teacher-trolling-by-proxy. All you need is an bright evil kid who can feign a look of innocent bewilderment. E.g., for algebra, "x^4+4 doesn't factor, does it?" Geometry: "How long are the two equal chords that trisect the area of the unit disk?" Or, "To construct the tangent to a circle at a given point, why are you showing us this complicated compass procedure with perpendicular bisectors, etc., when all you need is to add four more points at random, join all five in a star, draw two more lines, and you're done, no compass needed?" (See http://www.tweedledum.com/rwg/tan.htm .) Calculus: Which convergence test should I use for sum (csc n)/n^2 ? Trying to turn the latter into something tractable, try sum (csc π φ n)/n. Julian showed the largest terms have n = fib(k), and Csc[π*GoldenRatio*Fibonacci[k]]/Fibonacci[k] -> (-1)^Ceiling[k/3]*Sqrt[5]/π) so the partial sums of just the "spike" terms oscillate around 0 with period 6. But what about the exponentially many intervening small terms? Empirically, these follow suit, but now with three mysterious constants: Table[(Print[#]; #) &[ NSum[Csc[n*\[Pi]*GoldenRatio]/n, {n, Fibonacci[k] + 1, Fibonacci[k + 1] - 1}, WorkingPrecision -> 30, NSumTerms -> Fibonacci[k]]], {k, 34, 44}] 0.124338779160983782544979812601 -0.282342161451697923185165928059 0.0491007259722056399217080073056 -0.124338682934432043595276928760 0.282342101980413293488222974056 -0.0491007172954631024270905713056 0.124338705650441122683061852412 This last constant Shanks-extrapolates to ~ 1.243387013120 . Julian solved for the "spike terms" for the general quadratic integer. They all have periods dividing 6. --rwg Also, Julian challenges my assertion that you can improve long-term accuracy by *reducing* the minsky and cheb recurrence amplitudes. Stay tuned while I either demonstrate the effect or eat crow.