Let's say you find a determined bug - a bug with the singular ambition of walking in one direction. It covers no more and no less than one centimeter every second. You also have an unusual band a meter in circumference. You make a mark on it and start the bug at the mark. At the end of the first second you stretch the band so it adds another meter to its circumference, and so on... does the bug ever make it all the way around?
Most people guess that it can't - after all, the band is growing by a meter for every centimeter the bug travels. But let's analyze what's going on. The first insight is the band is expanding uniformly so it's expanding behind as well as in front of the bug. In the first second the bug travels one centimeter divided by one meter of the way around .. one percent. At the end of the second second it adds a centimeter, but the total distance is now two meters.. it has traveled an additional half percent of the total circumference, at the third second it has added another third of a percent, then a fourth and so on... If we add up the percentages we get a sum:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/n basically the sum of 1/n for all integers from n = 1 to infinity if the bug walks forever...
If you've had some physics or math, this is the harmonic series related to overtones of musical notes. Now on to the sum... does it converge to some number or is it unbounded running off to infinity?
Let's see if we can bound the series from above and below with series that are easier to calculate. Here's a really clever way to answer the question first proposed by the 13th century monk Nicolas Oresme. He made series where each element was less than or equal to each element in the harmonic series. By asking what is the largest power of one half that is less than or equal to the number you're looking at.
Consider the first term: the largest power of 1/2 that is less than or equal to 1 is (1/2)0 so the first term of the series is 1
the largest power of 1/2 that is less than or equal to 1/2 is (1/2)1 or 1/2 so the second term is 1/2
the largest power of 1/2 that is less than or equal to 1/3 is (1/2)2 or 1/4 .. the third term is 1/4
the larger power of 1/2 less than or equal to 1/4 is (1/2)2 or 1/4 .. the fourth term is 1/4
the largest power of 1/2 less than or equal to 1/5 is (1/2)3 or 1/8 .. the fifth term is 1/8
the largest power of 1/2 less than or equal to 1/6 is (1/2)3 or 1/8 .. our sixth term is 1/8
work out a few more for yourself and a pattern emerges
1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + ...
gathering the 1/2s
1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ... = 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 ...
1 + an infinite number of 1/2s equals infinity...
Every term in this series is less than or equal to each corresponding term of the harmonic series. Since it diverges, the larger harmonic series has to diverge. Given enough time our singularly determined bug only has to go 100 percent of the way around which is less than infinity so it eventually completes the trip. Working out how long it takes is a bit more involved, but a long time - about e100 seconds or about 1050 years..
This slow growth is logarithmic .. Using that as a hunch take the harmonic series to some number n and then subtract then natural log(n)
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/n - ln(n) .. or the [sum(1/n)] from 1 to n - ln(n)
as n goes to infinity we have an infinite number subtracted from an infinite number. The amazing thing is it converges to a rather small number. It's known as the Euler–Mascheroni constant and has been calculated to over 120 billion decimal digits .. the first few are: 0.5772156649...
It's called γ and turns up all over the place in physics and math. It's in quantum electrodynamics and the standard model - calculate corrections to the mass of the Higgs or the electron and there it is. eγ is used to explore products of prime numbers. It's not as famous as e, i, π, 1, 0, or ∞ .. but is one of the small list of important numbers used in our equations. Very little is known about it ... it is suspected to be transcendental, but that hasn't been proven. It isn't known if it is irrational.. There are other important numbers - fundamental constants for example, and the deep mystery of why math works so well in describing Nature. Way too deep for the blog, but perhaps something to frame up at some point.
Thai Carrot Soup
° tbl vegetable oil
° 1 medium onion roughly chopped
° 1 garlic clove chopped
° 1 stalk lemongrass stalk chopped
° 1 piece of ginger grated .. use your taste.. I did about an inch
° 1 pound of carrots peeled and chopped
° 1 six oz can of whole coconut milk
° 2-1/2 cups vegetable stock
° 1 lime zested and juiced
° heat the oil in a pan and cook garlic, onion, lemongrass and ginger for 4 or 5 minutes on medium high heat. Add carrots and go another 5
° reserve a few tbl of coconut milk and add the rest to the carrot mix. Simmer until carrots are soft
° throw in a blender or use a blender stick (I like the later) 'til smooth
° stir in lime juice and zest and then serve..
° drizzle with the left over coconut milk