To analyze the first series, let's break it down:

Part 1: Terms of the First Series

The first series is:

$S = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} + \ldots$

The pattern alternates between positive and negative terms, with each positive term followed by a negative term that matches its value but decreases by the next integer.

• 2n-1th term: This corresponds to the last positive term in the sequence up to $n$ pairs, which is:

  $a_{2n-1} = \frac{1}{n}$

• 2nth term: This corresponds to the last negative term in the sequence up to $n$ pairs, which is:

  $a_{2n} = -\frac{1}{n+1}$

Summary of Results

• 2n-1th term: $\frac{1}{n}$

• 2nth term: $-\frac{1}{n+1}$

Part 2: Convergence of the Series

1. First Series: The first series is an alternating series. By the Alternating Series Test, it converges since the absolute values of the terms decrease and approach zero.

2. Second Series: The second series is:

$T = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \ldots$

This series can be expressed in the form:

$T_n = \sum_{k=1}^{n} \frac{1}{k(k+1)} = \sum_{k=1}^{n} \left( \frac{1}{k} - \frac{1}{k+1} \right)$

By telescoping:

$T_n = 1 - \frac{1}{n+1} = \frac{n}{n+1}$

This series also converges as $n \to \infty$, converging to 1.

Part 3: Sum of the First $2n+1$ Terms of the First Series

To find the sum of the first $2n+1$ terms, we add the first $n$ positive terms and the first $n$ negative terms, and then add the additional positive term:

$S_{2n+1} = \left(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\right) - \left(\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\right) + \text{last term}$

Thus, the final sum can be expressed as:

$S_{2n+1} = 1 - \frac{1}{n+1}$

Conclusion

1. The series converge.
2. The sum of the first $2n+1$ terms is $1 - \frac{1}{n+1}$.

Follow-up Questions

1. How do we derive the general term for both series?
2. What is the explicit form of the sum for the first $n$ terms of the second series?
3. Can you illustrate the convergence behavior of both series graphically?
4. What happens if we extend the first series to infinity?
5. How do the two series relate to the harmonic series?

Tip

For alternating series, always check the convergence using the Alternating Series Test, which provides a useful criterion for convergence.