To express the series $1 - 4 + 16 - 64 + 256$ in sigma notation, we need to identify the pattern and general term of this sequence.

The sequence given is: $1, -4, 16, -64, 256$

Observing this, we can see:

1. Each term alternates in sign.
2. Each term is a power of 4: $1 = 4^0$, $-4 = -4^1$, $16 = 4^2$, $-64 = -4^3$, $256 = 4^4$.

Thus, a general term for this sequence could be: $(-1)^n \cdot 4^n$ where $n$ starts from $0$ and goes up to $4$.

The sigma notation for this series is therefore: $\sum_{n=0}^{4} (-1)^n \cdot 4^n$

Would you like further explanation on how the pattern was identified or any other details?

Here are some related questions you might explore:

1. How would the sigma notation change if the sequence started with a different power of 4?
2. What if the series had a different alternating sign pattern? How would that affect the general term?
3. Can this sigma notation be adapted if additional terms were added to this series?
4. How would the result change if the sign did not alternate?
5. What would happen if we changed the base of the powers from 4 to another number?

Tip: Always check for a pattern in both the powers and signs when expressing series in sigma notation.