sigma n=1 to infinity 1/(n+1)-1/n find n-th partial sum. Check its converge or diverge, if converge find its sum

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.

Learn how to calculate the n-th partial sum of the telescoping series sigma n=1 to infinity 1/(n+1)-1/n and determine if it converges or diverges. Discover the step-by-step solution, including identifying the series, writing out partial sums, and simplifying using telescoping nature. Suitable for advanced mathematics enthusiasts.

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