C. Write the following series in sigma notation and find their sum.
2-4/2+6/3-8/4+10/5

### Writing the Series in Sigma Notation

The given series is:
\[
2 - \frac{4}{2} + \frac{6}{3} - \frac{8}{4} + \frac{10}{5}
\]

To express this series in sigma notation, observe the pattern:

1. The numerators follow the sequence \(2, 4, 6, 8, 10\), which can be written as \(2n\), where \(n\) is the term number.
2. The denominators follow the sequence \(1, 2, 3, 4, 5\), which can be written as \(n\).
3. The series alternates in sign, starting with a positive term, which introduces a factor of \((-1)^{n+1}\) to alternate the signs correctly.

Thus, the series can be expressed in sigma notation as:
\[
\sum_{n=1}^{5} (-1)^{n+1} \cdot \frac{2n}{n}
\]

### Simplifying the Series

Let's simplify the expression inside the sigma notation:

\[
\sum_{n=1}^{5} (-1)^{n+1} \cdot \frac{2n}{n} = \sum_{n=1}^{5} (-1)^{n+1} \cdot 2
\]

This simplifies to:
\[
2\sum_{n=1}^{5} (-1)^{n+1}
\]

### Finding the Sum of the Series

Now, calculate the sum:

\[
S = 2\left[(-1)^{1+1} + (-1)^{2+1} + (-1)^{3+1} + (-1)^{4+1} + (-1)^{5+1}\right]
\]

\[
S = 2\left[1 - 1 + 1 - 1 + 1\right]
\]

Adding up the terms inside the brackets:

\[
S = 2 \times 1 = 2
\]

So, the sum of the series is **2**.

---

Do you want more details or have any questions? 

Here are some related questions:

1. How would the series change if the pattern of signs was different?
2. Can the series be extended beyond 5 terms? What would be its sum?
3. What if the numerators followed a different arithmetic sequence?
4. How does the alternating sign affect the sum?
5. How would you express the series if it started with a negative term?

**Tip:** When dealing with alternating series, always check how the signs affect the sum before simplifying.

Explore how to write a series in sigma notation and find its sum with an example problem. This problem involves alternating signs and arithmetic sequences of numerators and denominators. Suitable for advanced high school students.

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