<s>[INST] 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
 [/INST] The series is similar to $\sum \limits_{n=3}^{\infty} \frac{1}{2^n}=2-1/1-1/2-1/4=1/4$.
Note that $\sum \limits_{n=0}^{\infty} x^n=\frac{1}{1-x}$ for $|x|<1$.
EDIT, since I misread the question:
$$1/8 - 1/16 + 1/32 + 1/64 - 1/128 - 1/256 + 1/512 + 1/1024 - 1/2048 - 
1/4096 + 1/8192 + 1/16384 - 1/32768 - 1
/65536 + ...=$$
$$\frac{1}{8}-\frac{1}{16}+\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{2^{2n+1}}+\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{2^{2n+2}}=$$
$$\frac{1}{8}-\frac{1}{16}+\frac{1}{40}+\frac{1}{80}=\frac{1}{10}$$
This expression can be evaluated using the formula given above
</s>