We are given the infinite series:

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

To determine whether this series converges or diverges, we will use the concept of partial sums and analyze the behavior of the series. First, let's express the general term of the series:

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

Step 1: Behavior of the General Term

As $n \to \infty$, let's analyze the limit of the general term $a_n = \frac{n}{n+2}$:

$\lim_{n \to \infty} \frac{n}{n+2} = \lim_{n \to \infty} \frac{n}{n(1 + \frac{2}{n})} = \frac{1}{1 + 0} = 1$

Since the general term $a_n$ does not approach zero but instead approaches 1, the series cannot converge. For a series to converge, the general term must tend to zero as $n \to \infty$.

Thus, the series diverges by the Test for Divergence (also known as the nth-term test for divergence).

Conclusion:

The series $\sum_{n=1}^{\infty} \frac{n}{n+2}$ diverges.

Would you like more details on divergence tests or any related topic?

5 Related Questions:

1. What is the nth-term test for divergence?
2. How does the partial sum approach help in determining convergence?
3. Can you explain the difference between conditional and absolute convergence?
4. What are some common tests for series convergence, such as the ratio or root test?
5. How can we determine the sum of a convergent geometric series?

Tip:

For a series to converge, the terms must approach zero as $n \to \infty$; otherwise, the series will diverge. Always check the limit of the general term first!