We are tasked with evaluating the limit:

$\lim_{n \to \infty} \left( \frac{1 + 3 + 5 + \cdots + (2n - 1)}{n + 1} - \frac{2n + 1}{2} \right)$

Step 1: Simplifying the Sum of Odd Numbers

The sum of the first $n$ odd numbers $1 + 3 + 5 + \cdots + (2n - 1)$ is a well-known result:

$1 + 3 + 5 + \cdots + (2n - 1) = n^2$

Thus, we can replace the sum in the expression:

$\lim_{n \to \infty} \left( \frac{n^2}{n + 1} - \frac{2n + 1}{2} \right)$

Step 2: Simplifying Each Term

Now, simplify the two terms separately:

1. $\frac{n^2}{n+1}$:

$\frac{n^2}{n+1} = \frac{n^2}{n(1 + \frac{1}{n})} = \frac{n}{1 + \frac{1}{n}} = \frac{n}{1 + \frac{1}{n}} \to n \text{ as } n \to \infty$

2. $\frac{2n + 1}{2}$:

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

Step 3: Subtract the Two Expressions

Now subtract the two simplified terms:

$\lim_{n \to \infty} \left( n - \left( n + \frac{1}{2} \right) \right)$

This simplifies to:

$\lim_{n \to \infty} \left( n - n - \frac{1}{2} \right) = \lim_{n \to \infty} -\frac{1}{2} = -\frac{1}{2}$

Final Answer:

$\boxed{-\frac{1}{2}}$

Would you like further clarification on any steps? Here are some related questions:

1. How can we generalize sums of odd numbers?
2. What is the limit of rational functions as $n \to \infty$?
3. How do we apply the squeeze theorem to evaluate limits?
4. What other series converge in a similar manner?
5. What are other common sums involving consecutive odd or even numbers?

Tip: When dealing with limits at infinity, try to simplify each term by factoring out dominant terms for easier analysis.