A cute and important trick is based on "sliding things down a notch." Consider the decimal number 0.12121212... repeated forever. What fraction is it? Call it X; multiply by 100 to get 12.121212... and subtract the original; all the infinity of decimal places cancel out leaving only 12 behind. So 100*X- X = 12 — that is, 99*X = 12 and so X must be 12/99 (= 4/33).
The same principle applies in a host of other contexts. IF a function can be written in powers of X (a series expansion) then dividing by X slides everything down a notch. With some luck and ingenuity one can then engineer a massive cancellation and reduce the problem to a simpler relationship. Analogous patterns hold for many continued fractions, differential equations, and so on.
Outside of mathematics the cancellation isn't likely to be perfect, but it still can be a helpful tool for analysis. Crystals are built of (nearly) regular periodic lattices of atoms, and sliding things down a notch helps to explain all sorts of solid-state physical phenomena such as sound, heat, and electronic properties. Generations of plants and animals (and people) succeed one another, and some insights may be found by looking at cross-generational differences. Perhaps similar tactics can aid thinking about problems in economics, politics, and philosophy?
Saturday, February 26, 2000 at 07:12:39 (EST) = 2000-02-26
When dealing with infinite sums or series, this same event can occur. When you add terms, you find that alternate terms cancel, leaving only a few terms from the beginning.
One such telescoping series is the sum of 1 / (4n^2 - 1).
-RadRob
(correlates: AfterlifeGrosses, Comments on HereBeDragons, GrayGreenGap, ...)