One of my favorite mathematical theorems that I've been exposed to so far is the divergence of the sum of the prime reciprocals (as referenced here and proven here). It says that if you take all the prime numbers (which are infinite) and turn them each upside down so that they're in the denominator of a fraction, and add up all those fractions, the resulting sum will be infinite. If this doesn't sound remarkable, let's try a few.
The first ten primes:
1/2 0.50000000000000000000 1/3 0.33333333333333333333 1/5 0.20000000000000000000 1/7 0.14285714285714285714 1/11 0.09090909090909090909 1/13 0.07692307692307692307 1/17 0.05882352941176470588 1/19 0.05263157894736842105 1/23 0.04347826086956521739 1/29 0.03448275862068965517 Total= 1.53343877187203202214
Not very big, but of course ten primes is not that many either. Let's try some more
The first 500 primes, 2 --> 3571
... ....... 1/3541 0.00028240609997175939 1/3547 0.00028192839018889202 1/3557 0.00028113578858588698 1/3559 0.00028097780275358246 1/3571 0.00028003360403248389 Total= 2.36532946588456968519
By adding 490 more primes, we increased our total sum from 1.533 to 2.365. Finally, if we add the reciprocals of the first million primes, the result is only 3.06821904805 (that's as far as my computer is interested in calculating). I doubt it would ever count high enough to get to 4. And yet, the theorem says that if the process continues indefinitely, the sum will get as high as you like.
I like it.
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