Zeno Hears the Music!

As you may recall from the last post, Zeno was left frustrated by never getting that prairie cleared of invasive brush. He had followed the strategy of cutting and treating in a way that would eliminate half of the brush each year, leaving half to reemerge as resprouts. He thought that this would be good enough to eventually get rid of the brush and didn't mind missing a few side branches, cutting the stumps high, or not apply enough herbicide to the stump as long as the goal of getting rid of half of what was there each year was achieved. But, as time went on, he found that he could never seem to get the prairie completely free of brush. He found it was like trying to cross a road by first going half way across, stopping for a rest, then going half of the remaining distance, resting again and doing this over and over, but never reaching his goal. Similarly, Zeno found that by following this strategy on the prairie, he never got it completely cleared of brush. Rats!

So, as Zeno was pondering about restoring his next prairie, he thought it over while listening to some music. Being a musician, he remembered that music followed a "harmonic series". That is, unlike the series above which can be written, 1/2 , 1/4, 1/8, 1/16..., the harmonic series was different. It followed a progression that is more pleasing to the ear and looks like this- 1/2, 1/3, 1/4, 1/5, 1/6. And then it dawned on Zeno that the answer to his dilemma of never completely restoring his prairie was to "hear the music" and follow the harmonics!

Here was Zeno's thinking- if he started with 1000 stems of buckthorn on the prairie, following his previous strategy of eliminating half of that remaining each year (following the series of 1/2, 1/4, 1/8,... of 1000) he would be down to 500 plants after one year, then 250, then 125, then 62.5, then 32...and never reach zero. That stragegy followed a "convergent series" approaching but never reaching a finite limit. In contrast, if he followed the harmonic series, a "divergent series", one that never stops growing and, indeed, approaches infinity, he would eliminate 500 plants the first year, then 1/3 of 1000 or 333 plants the next year, then 1/4 of 1000 or 250 plants on year 3 which would put him over the 1000. Hooray, he would have reached his goal in three short years!

So, from then on, Zeno, and all of his volunteers, renewed their efforts to be meticulous in their restoration efforts by carefully cutting all the stems and side branches close to the ground and applying adequate herbicide to each stump, following a "harmonious" path that would eventually get rid of that pesky brush... and turn something like this:

into this: