📚 Topic Summary
Telescoping series are series where most of the terms cancel out, leaving only a few terms at the beginning and end. This 'cancellation' makes it easier to find the sum of the series. Identifying this pattern is key. Often you'll need to use partial fraction decomposition to see how the terms break down and cancel.
The general idea is to express each term $a_n$ as a difference $b_n - b_{n+1}$ or $b_{n+1} - b_n$. When you add up many of these terms, most of the $b_n$ values will disappear, leaving only a few.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Partial Sum |
A. The sum of an infinite series that approaches a finite value. |
| 2. Telescoping Series |
B. A sequence of sums obtained by adding the first $n$ terms of a series. |
| 3. Convergence |
C. A series where internal terms cancel, leaving a finite number of terms. |
| 4. Divergence |
D. The series does not approach any finite value. |
| 5. Partial Fraction Decomposition |
E. A method to rewrite rational functions into simpler fractions. |
✍️ Part B: Fill in the Blanks
A telescoping series is a series where most of the ______ cancel. To determine the sum of a telescoping series, find the ______ sum and evaluate the limit as n approaches ______. If the limit exists, the series ______. Otherwise, it ______.
🤔 Part C: Critical Thinking
Explain in your own words how partial fraction decomposition helps in evaluating telescoping series. Provide an example.