To find the partial sum of the telescoping series

$S = \sum_{n=5}^{\infty} \left(\frac{1}{n+1} - \frac{1}{n+2}\right),$

let's analyze the expression step by step.

Step 1: Simplifying the General Term

The general term of the series is:

$\frac{1}{n+1} - \frac{1}{n+2}.$

This is a telescoping series, meaning that many terms will cancel out when summed up.

Step 2: Writing the First Few Terms

The series starts at $n = 5$ and goes to infinity. Let's expand the first few terms:

$S = \left(\frac{1}{6} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{8}\right) + \left(\frac{1}{8} - \frac{1}{9}\right) + \cdots$

Step 3: Telescoping Effect

Notice that many terms cancel out:

• $\frac{1}{7}$ cancels with $-\frac{1}{7}$,
• $\frac{1}{8}$ cancels with $-\frac{1}{8}$,
• and so on.

The cancellation will leave only the first part of the first term and the second part of the last term (as the number of terms tends to infinity).

Thus, after cancellation, we are left with:

$S = \frac{1}{6}.$

Conclusion

The partial sum of the given telescoping series is:

$S = \frac{1}{6}.$

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Would you like further details or have any questions about this process?

5 Relative Questions:

1. What are some other examples of telescoping series?
2. How do you determine if a series will telescope?
3. Can a telescoping series converge to values other than zero?
4. What happens if the starting index of the series changes?
5. How does the convergence of a telescoping series compare to a geometric series?

Tip:

In telescoping series, always look for terms that will cancel out, leaving only a few terms from the beginning and end.