C. Write the following series in sigma notation and find their sum.
5+10/3+15/5+20/7+25/9

### Writing the Series in Sigma Notation

The given series is:
\[
5 + \frac{10}{3} + \frac{15}{5} + \frac{20}{7} + \frac{25}{9}
\]

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

1. The numerator of each term increases by 5: \(5, 10, 15, 20, 25\), which can be written as \(5n\), where \(n\) is the term number.
2. The denominator of each term starts at 1 and increases by 2 for each subsequent term: \(1, 3, 5, 7, 9\), which can be written as \(2n - 1\).

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

### Finding the Sum of the Series

Now, let's calculate the sum of the series:

\[
S = \sum_{n=1}^{5} \frac{5n}{2n-1}
\]

Substituting \(n = 1\) to \(n = 5\):

\[
S = \frac{5(1)}{2(1)-1} + \frac{5(2)}{2(2)-1} + \frac{5(3)}{2(3)-1} + \frac{5(4)}{2(4)-1} + \frac{5(5)}{2(5)-1}
\]
\[
S = \frac{5 \times 1}{1} + \frac{5 \times 2}{3} + \frac{5 \times 3}{5} + \frac{5 \times 4}{7} + \frac{5 \times 5}{9}
\]
\[
S = 5 + \frac{10}{3} + \frac{15}{5} + \frac{20}{7} + \frac{25}{9}
\]

Now let's calculate each term individually:

\[
S = 5 + 3.333 + 3 + 2.857 + 2.778
\]
\[
S = 17.968
\]

So, the sum of the series is approximately **17.968**.

---

Do you want more details or have any questions? 

Here are some related questions:

1. How can the sigma notation be extended for more terms in the series?
2. What is the general formula for summing such series?
3. How can you find the exact sum without approximations?
4. What if the denominator increased by a different constant instead of 2?
5. How does this series relate to harmonic series?

**Tip:** When working with fractions in series, it's often useful to look for patterns in both the numerators and denominators.

Explore how to express and compute the sum of the series 5 + 10/3 + 15/5 + 20/7 + 25/9 in sigma notation. Learn the pattern in the numerator and denominator of the series. Calculate the sum step-by-step with detailed explanations.

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