I'm trying to solve for this summation:
$$\sum_{j=0}^{i} {\left(\frac 1 2\right)^j}$$
This looks a lot like a geometric series, but it appears to be inverted. Upon plugging the sum into Wolfram Alpha, I find the answer to be
$2-2^{-i}$
but I don't understand how it gets there. Am I able to consider this a geometric series at all? It almost seems closer to the harmonic series.
Best Answer
It is a geometric series:
$$\sum_{j=0}^{i} \frac{1}{2^j}=\sum_{j=0}^{i}\left( \frac{1}{2}\right)^j=\frac{1-\left(\frac12\right)^{i+1}}{1-\frac12}=2\left(1-\left(\frac12\right)^{i+1}\right)=2-\left(\frac12\right)^{i}=2-2^{-i}$$