An exponent of a whole number is like one with a fraction.................after 1/2^x is 1/3^x................b/c 1/3 cannot be a part of the halving series....................1/2 to the x........the den................the number on the bottom is like 2 to the x....................2^............the different exponents with 2 as a base.............after 1/3 is 1/5...........b/c 1/4 is inside of 1/2 as a geo series..................1/6 is after 1/5..........none of the previous fractions would contain 1/6......................the halving series goes..............1/2, 1/4, 1/8, 1/16, 1/32, 1/64........................which seems strange b/c in 1/6............6 is not prime.............................but 1/3 or 1/5 does not contain 1/6 with their geo series...............so from 1/10 and below..............there are 6 fractions...........that would start their own geo series..........hence my language........"original" geo series...............1/9 is contained in 1/3's geo series..............so the 3 exceptions..................1/4, 1/8, 1/9 are from 1/2 and 1/3................the only two geo series of the 1st ten...........................the ones under 1/10 to contain..........other fractions in that set of fractions..........imp. b/c 2 and 3 start the primes and are the only 2 primes but a digit apart...........only 2 primes a digit apart................like a line at 1/2........................



I am particularly concerned about these two sums:

∑x=0∞12x=1+12+14...=2$$\sum _{x=0}^{\mathrm{\infty }}\frac{1}{2^x}=1+\frac{1}{2}+\frac{1}{4}...=2$$

and

∑x=1∞1x=1+12+13...=∞