Final answer:
The series can be expressed in summation notation as \(\sum_{n=1}^{N} (-1)^{n+1} \frac{1}{2n}\), where N is the number of terms in the series.
Step-by-step explanation:
The student has provided a pattern of terms that alternates between negative and positive fractions with even denominators. The pattern begins with -(1/2), followed by (1/4), -(1/6), (1/8), and so on. We can summarize this pattern using summation notation which compactly represents the sum of terms in a series.
To write the series in summation notation, we identify a general term that represents each term in the series. Observing the pattern, the numerator is always 1, and the denominator is an increasing even number, which we can represent as 2n. Since the series alternates in sign, we use (-1)^{n+1} to alternate signs for each term. The corresponding summation notation for the given series is:
Σ ( -1)n-1(1/(2n)(2n+2))
where Σ denotes the summation sign, n is the index of summation, and the terms inside the parentheses represent the pattern of the series.
Therefore, the general term of the series can be written as (-1)^{n+1} * (1/(2n)). Now, if we want this to start with n=1 for our first term -(1/2), the summation would look like:
\[∑_{n=1}^{N} (-1)^{n+1} \frac{1}{2n}\]
Here, N represents the number of terms to sum. Without knowing the specific number of terms, N is left as a placeholder.