To determine whether the given infinite series converges, we need to analyze its behavior. Let’s carefully break it down.

The series is:

$\sum_{n=1}^{\infty} \left( \frac{n}{n+1} - \frac{n+2}{n+3} \right)$

Step 1: Simplify the general term

The $n$-th term of the series is: $a_n = \frac{n}{n+1} - \frac{n+2}{n+3}.$

Let’s simplify each fraction:

1. For $\frac{n}{n+1}$: $\frac{n}{n+1} = 1 - \frac{1}{n+1}.$

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

Now substitute back into $a_n$: $a_n = \left(1 - \frac{1}{n+1}\right) - \left(1 - \frac{1}{n+3}\right).$

Simplify: $a_n = -\frac{1}{n+1} + \frac{1}{n+3}.$

Thus, the series becomes: $\sum_{n=1}^\infty a_n = \sum_{n=1}^\infty \left(-\frac{1}{n+1} + \frac{1}{n+3}\right).$

Step 2: Check for telescoping

Notice that the terms in the series "telescope." Write out the first few terms explicitly: $a_1 = -\frac{1}{2} + \frac{1}{4},$ $a_2 = -\frac{1}{3} + \frac{1}{5},$ $a_3 = -\frac{1}{4} + \frac{1}{6},$ and so on.

When summed together, most terms cancel out, leaving only a few terms at the start and the end: $S_N = \text{(remaining terms after cancellation)}.$

Step 3: Express the partial sum

Let’s calculate the partial sum $S_N$: $S_N = -\frac{1}{2} - \frac{1}{3} + \frac{1}{N+2} + \frac{1}{N+3}.$

Step 4: Take the limit as $N \to \infty$

Now, analyze the behavior of $S_N$ as $N \to \infty$:

1. The terms $\frac{1}{N+2}$ and $\frac{1}{N+3}$ approach 0 as $N \to \infty$.
2. The terms $-\frac{1}{2} - \frac{1}{3}$ remain constant.

Thus, the sum converges to: $S = -\frac{1}{2} - \frac{1}{3} = -\frac{5}{6}.$

Final Answer:

The series converges, and its sum is: $\boxed{-\frac{5}{6}}.$

Do you want further clarification or details?
Here are 5 related questions to deepen understanding:

1. What is the general test for telescoping series convergence?
2. How does term cancellation help simplify infinite series?
3. What happens if the telescoping series does not simplify completely?
4. How does this result compare to the harmonic series?
5. Can we use other tests (e.g., comparison test) to verify convergence?

Tip: Always look for patterns of cancellation in series with terms involving differences! It simplifies the analysis.