I just need to know how I would go about evaluating this:
$$ \sum_{t=0}^{n-1} v^t\frac{(1-v^t)}{i} = \frac{1}{i}\sum_{t=0}^{n-1} v^t(1-v^t) $$
$$ =\frac{1}{i}\sum_{t=0}^{n-1} v^t - \frac{1}{i}\sum_{t=0}^{n-1}v^{2t} $$
How would I evaluate this? I really cannot seem to figure out how the Riemann sums would look for both t and 2t as the exponent.
First (geometric) sum is $\frac{1-v^n}{1-v}$ Your answer is correct only if $|v|\lt 1$ and upper limit is $\infty$
Second sum is also geometric $=\frac{1-v^{2n}}{1-v^2}$
The difference (for $|v|\lt 1$ and $n\to \infty$) is $\frac{v}{1-v}$ not $1$, unless $v=0.5$.