Analyzing the Convergence of the Series \( \sum_{n=1}^\infty \frac{2n + 1}{n} \)

We are asked to determine whether the series

\[ \sum_{n=1}^\infty \frac{2n + 1}{n} \]

converges or diverges.

Let’s begin by examining the general term of the series:

\[ a_n = \frac{2n + 1}{n} \]

We can simplify the expression as follows:

\[ a_n = \frac{2n}{n} + \frac{1}{n} = 2 + \frac{1}{n} \]

So each term of the series takes the form:

\[ a_n = 2 + \frac{1}{n} \]

To assess convergence, we consider the limit of the general term as \( n \to \infty \). A necessary condition for the convergence of an infinite series \( \sum a_n \) is that its general term tends to zero:

\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(2 + \frac{1}{n} \right) = 2 \]

Since the general term does not approach zero, but instead tends to 2, this condition is not satisfied:

\[ \lim_{n \to \infty} a_n = 2 \ne 0 \]

Therefore, the series diverges by the divergence (or nth-term) test.

And so on.