This image appears to be an explanation about the divergence of a series. Let’s break it down.

The series in question is:

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

Step-by-Step Explanation:

1. Expression of the General Term: The term inside the summation is $\frac{n+1}{n}$. This can be simplified as:

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

   So the general term is the sum of two components: 1 and $\frac{1}{n}$.

2. Analyzing the Limit: The series diverges because the term $\frac{n+1}{n}$ does not approach zero as $n \to \infty$. Instead, we calculate the limit of the general term as $n \to \infty$:

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

3. Test for Divergence: For a series $\sum a_n$ to converge, a necessary condition is that:

   $\lim_{n \to \infty} a_n = 0$

   In this case, since the limit of the general term is 1, the series fails this test, and therefore, the series diverges.

Conclusion:

The series diverges because the general term does not approach zero but instead tends to 1. Since the limit of the general term does not go to zero, we can conclude that the series cannot converge.

Would you like further clarification or a more in-depth explanation?

Here are 5 related questions you could explore:

1. What is the test for convergence of a series?
2. How does the comparison test work in determining the convergence of a series?
3. What is the difference between an absolutely convergent and conditionally convergent series?
4. How do you apply the ratio test to a series?
5. What happens when the limit of a series' terms is greater than 0?

Tip: Always check the limit of the general term of a series when determining convergence or divergence. If it doesn’t approach zero, the series cannot converge.