0.999.../The geometric series formula


 * $$\sum_{n=0}^\infty x^n = 1 +x +x^2 + x^3 + ...$$ is known as a geometric series.

If, $$|x|<1$$, then this series converges to:


 * $$\sum_{n=0}^\infty x^n= \frac{1}{1-x} $$

Proof: Define the partial sum $$S_n$$:
 * $$S_n = \sum_{n=0}^n = 1 + x + x^2 + x^3 + ... x^{n-1}$$
 * $$xS_n = \sum_{n=0}^n = .. + x + x^2 + x^3 + ... x^{n-1} + x^n$$

Note that both partial sums have n terms. When they are subtracted only the first term in and the last term in $$xS_n$$ will remain:


 * $$S_n -x S_n = 1-x^n$$, which can be easily solved for $$S_n$$

Note that we have failed to establish when the infinite series converges. This requires an understanding of the what happens when we take the limit of the partial sum as n goes to infinity. This is left to the reader as problem 1.

Sample problems:


 * 1) Prove that sequence of partial sums, $$\{S_n\}_{n=1}^{\infty} = \{S_1, S_2, S_3, ...\} $$, converges if $$|x|<1$$.
 * 2) This can be used to convert other repeating decimals into fractions. For pedagogical purposes, it is better to begin with a known fraction. From such a list, we consider: 0.181818... . Convert this into a rational fraction.