The powers of \( i \) exhibit a repeating cycle of length 4: \(i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1\). This pattern repeats.
The sum \( i^2 + i^3 + \dots + i^{4000} \) can be viewed as a sequence of these cycles:
\[
(i^2 + i^3 + i^4 + i^1) + (i^2 + i^3 + i^4 + i^1) + \dots \text{(1000 repetitions)}.
\]
Each cycle sums to:
\[
i^2 + i^3 + i^4 + i^1 = -1 - i + 1 + i = 0.
\]
Therefore, the sum of these complete cycles is \(0 \cdot 1000 = 0\).
The remaining terms are \( i^2 \) and \( i^3 \):
\[
i^2 + i^3 = -1 + i.
\]
Consequently, the total sum evaluates to:
\[
0 + (-1 + i) = -1 + i.
\]
The final result is:
\[
\boxed{-1 + i}.
\]