We are tasked with finding the 5th partial sum, $S_5$, of the given alternating series:

$S = \sum_{n=1}^{\infty} \frac{(-1)^n}{2n+1}$

The 5th partial sum is defined as:

$S_5 = \sum_{n=1}^{5} \frac{(-1)^n}{2n+1}$

To find $S_5$, we will compute the sum for the first five terms of the series.

Step-by-Step Calculation:

1. For $n = 1$: $\frac{(-1)^1}{2(1) + 1} = \frac{-1}{3} = -\frac{1}{3}$

2. For $n = 2$: $\frac{(-1)^2}{2(2) + 1} = \frac{1}{5}$

3. For $n = 3$: $\frac{(-1)^3}{2(3) + 1} = \frac{-1}{7}$

4. For $n = 4$: $\frac{(-1)^4}{2(4) + 1} = \frac{1}{9}$

5. For $n = 5$: $\frac{(-1)^5}{2(5) + 1} = \frac{-1}{11}$

Now, sum these values to find $S_5$:

$S_5 = -\frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11}$

Let me calculate this sum for you.The 5th partial sum, $S_5$, is:

$S_5 = -\frac{887}{3465} \approx -0.256$

This is the exact value of the sum in fraction form, and the approximate decimal value is $-0.256$.

Do you have any questions, or would you like more details on this?

Here are some related questions to explore further:

1. What is the behavior of the infinite sum of this series? Does it converge?
2. Can we apply the alternating series test to determine if this series converges?
3. How would you calculate the 10th partial sum of this series?
4. What are the criteria for the convergence of an alternating series?
5. How would the series look if we altered the denominator to $2n + 2$?

Tip: For alternating series, understanding the behavior of partial sums can give insights into the overall convergence of the series.