The given series is:

$1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \cdots + 597 + 598 - 599 - 600$

We observe a repeating pattern of four terms:
$(+1, +2, -3, -4), (+5, +6, -7, -8), \cdots$

This pattern repeats every four terms, so the entire sum can be divided into blocks of four terms.

Grouping the terms:

Let's group the terms as: $(1 + 2 - 3 - 4), (5 + 6 - 7 - 8), \cdots$

Each group sums up to: $(1 + 2 - 3 - 4) = -4, (5 + 6 - 7 - 8) = -4, \cdots$ Each group of four terms contributes $-4$.

Now, let's determine how many such groups we have. There are 600 terms in total, and since each group consists of 4 terms, the number of groups is: $\frac{600}{4} = 150$

Calculating the total sum:

Each group contributes $-4$, and there are 150 groups. Therefore, the total sum is: $150 \times (-4) = -600$

Thus, the value of the series is:

$\boxed{-600}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the sum of the series $1 - 2 + 3 - 4 + 5 - 6 + \cdots$?
2. How do you find the sum of an alternating series?
3. How does the grouping method help in solving long series?
4. Can this pattern method be used for non-alternating arithmetic series?
5. How would the result change if the series included only positive terms?

Tip: Always check for repeating patterns when dealing with long arithmetic or alternating series. It simplifies the calculation significantly.