This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

$\large \dpi{200} \bg_white f^0+f^1+f^2+f^3+f^4+f^5+f^6+ \ldots = {1 \over 1-f}$

Or, in fancy notation:

$\large \dpi{200} \bg_white \sum_{i=0}^\infty f^{\,i} = {1 \over 1-f} , \quad 0

Or, in C++:

long big = 9999999999;float frac = .70;double total = 0;for ( i = 0; i < big; i++){  total += frac∗∗i;  }cout << total;                 # in this case, prints 1 / .3 = 10/3cout << total - 1/(1-frac)     # prints 0 for any value of frac

Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

$\large \dpi{200} \bg_white {1 \over 2} + {1 \over 4} +{1 \over 8} + {1 \over 16 } + {1 \over 32} + {1 \over 64} + {1 \over 128} + {1 \over 256} + \ldots = {^1\! / \!_2 \over 1- {1 \over 2} } = 1$

Take that, Zeno!