The point..........my point is this............it is like exponents..............................square roots..........1/2............cube roots..........and exponents.........like 2 squared or 2 cubed are reverse processes of each other....

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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...=∞$$\sum _{x=1}^{\mathrm{\infty }}\frac{1}{x}=1+\frac{1}{2}+\frac{1}{3}...=\mathrm{\infty }$$ Now, I know and I understand the proofs that say the first sum converges to two and the second one is divergent. ( you probably shouldn't equate a divergent series like that ). But on an intuitive level, what separates these two sums in such a way, that their limits are so drastically different? Both sums add very small numbers in the end ( e.g. 110000000000$\frac{1}{10000000000}$ ), yet one always stays smaller than two and one goes way beyond that. You would think ( again, on an intuitive level, the math is clear) that the numbers become so small that there is a certain number that doesn't change anymore (significantly). For example: ∑x=11001x=ca.5$$\sum _{x=1}^{100}\frac{1}{x}=ca.5$$