To find the $n$-th partial sum of the series $\sum_{n=1}^{\infty} \left(\frac{1}{n+1} - \frac{1}{n}\right)$ and check if it converges or diverges, follow these steps:

1. Identify the series: $\sum_{n=1}^{\infty} \left(\frac{1}{n+1} - \frac{1}{n}\right)$

2. Express the $n$-th partial sum $S_N$: $S_N = \sum_{n=1}^{N} \left(\frac{1}{n+1} - \frac{1}{n}\right)$

3. Write out the partial sums to recognize a pattern: $S_N = \left(\frac{1}{2} - \frac{1}{1}\right) + \left(\frac{1}{3} - \frac{1}{2}\right) + \left(\frac{1}{4} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{N+1} - \frac{1}{N}\right)$

4. Simplify the sum using telescoping nature: Notice that most terms cancel out: $S_N = \left(- \frac{1}{1} + \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{1}{3}\right) + \left(- \frac{1}{3} + \frac{1}{4}\right) + \cdots + \left(- \frac{1}{N} + \frac{1}{N+1}\right)$ Simplifies to: $S_N = -1 + \frac{1}{N+1}$

5. Take the limit as $N$ approaches infinity to determine convergence: $\lim_{N \to \infty} S_N = \lim_{N \to \infty} \left(-1 + \frac{1}{N+1}\right) = -1 + 0 = -1$

So, the series converges and its sum is $-1$.

Summary:

• The series converges.
• The sum of the series is $-1$.

Would you like a detailed step-by-step explanation of how we reached this conclusion or do you have any specific questions?

Here are some related questions to expand your understanding:

1. What is a telescoping series?
2. How do you determine if a series is telescoping?
3. What other types of series are there besides telescoping?
4. What is the importance of finding the partial sum of a series?
5. How can you determine if a series converges or diverges in general?
6. What are some real-world applications of infinite series?
7. How does the comparison test work for determining convergence?
8. What is the difference between an arithmetic series and a geometric series?

Tip: When working with series, always look for patterns like telescoping, which can simplify the process of finding sums and determining convergence.