Infinity Series to Approximate a fraction I have observed that the following series is a good approximation for
$\frac{1}{10}$.
$\frac{1}{8}- \frac{1}{16} + \frac{1}{32} + \frac{1}{64} - \frac{1}{128} - \frac{1}{256} + \frac{1}{512} + \frac{1}{1024} - \frac{1}{2048} - 
\frac{1}{(2\cdot2048)} + \frac{1}{(4\cdot2048)} + \frac{1}{(8\cdot2048)} - \frac{1}{(16\cdot2048)} - \frac{1}{(32\cdot2048)} + ...$
I believe there is a pattern to the series. However, I cannot prove that pattern holds. Does it? Can it be proved? I am thinking some Taylor series might be the way to go but that is just a hunch.
Bob

$$\frac{1}{8}-\frac{1}{16}+\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{2^{2n+1}}+\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{2^{2n+2}}=\dots$$
