To express the given sum in sigma notation, let's analyze the series:

Given series:

$\frac{2}{6} - \frac{2}{9} + \frac{2}{12} - \frac{2}{15} + \dots + \frac{2}{24}$

Step-by-step breakdown:

1. Numerator: The numerator is constant and equal to $2$.

2. Denominator: The denominators form an arithmetic sequence: $6, 9, 12, 15, \dots, 24$ The sequence starts at $6$, and the common difference is $3$.

   Denominator for the $n$-th term: $a_n = 6 + 3(n-1) = 3n + 3$

3. Signs: The series alternates between positive and negative, starting with a positive term. This alternation is modeled using $(-1)^{n+1}$.

4. General term: Combining these, the general term for the series is: $\frac{2 \cdot (-1)^{n+1}}{3n + 3}$

5. Sigma notation: The series can be expressed as: $\sum_{n=1}^{7} \frac{2 \cdot (-1)^{n+1}}{3n + 3}$

Here, the upper limit $7$ corresponds to the last term $24$ in the denominator.

Would you like further clarification or assistance with expanding the series?

Related Questions:

1. How can the sum of the series be calculated explicitly?
2. What is the role of alternating signs in the series?
3. How do you find the general term for a given sequence?
4. How do arithmetic sequences relate to sigma notation?
5. How would this series look if the numerator also varied?

Tip: Always verify the pattern of the sequence (numerator, denominator, and signs) before attempting sigma notation!