Solved Problems: Telescoping Series Calculus Examples

📚 Quick Study Guide

• 🔍Definition: A telescoping series is a series where most terms cancel out, leaving only a few terms. This makes it possible to find the sum of the series.
• 🔢General Form: Often, a telescoping series can be written in the form $\sum_{n=1}^{\infty} (b_n - b_{n+k})$ for some constant $k$.
• 💡Partial Fractions: This is a common technique used to rewrite the terms of the series into a difference that leads to telescoping.
• 📝Partial Sums: Compute the partial sums $S_N = \sum_{n=1}^{N} (b_n - b_{n+1})$ and observe the cancellation pattern.
• ✔️Limit: If $\lim_{N \to \infty} S_N$ exists, then the telescoping series converges to that limit. If the limit does not exist, the series diverges.

Practice Quiz

1. Question 1: Which of the following series is a telescoping series?
   1. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   3. $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$
   4. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
2. Question 2: What is the first step in evaluating a telescoping series?
   1. Find the limit of the terms.
   2. Compute the partial sums.
   3. Apply the ratio test.
   4. Check for divergence.
3. Question 3: Evaluate the telescoping series $\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)$.
   1. 0
   2. 1
   3. $\infty$
   4. -1
4. Question 4: What happens to the terms in a telescoping series?
   1. They all become zero.
   2. Most terms cancel out.
   3. They increase exponentially.
   4. They alternate in sign.
5. Question 5: Which technique is commonly used to transform a series into a telescoping series?
   1. Integration by parts
   2. U-substitution
   3. Partial fractions
   4. L'Hôpital's Rule
6. Question 6: Determine the sum of the series $\sum_{n=1}^{\infty} \left(\frac{1}{n+2} - \frac{1}{n+3}\right)$.
   1. $\frac{1}{2}$
   2. $\frac{1}{3}$
   3. $\frac{1}{5}$
   4. 1
7. Question 7: What is the limit that needs to be evaluated to find the sum of a telescoping series?
   1. The limit of the individual terms.
   2. The limit of the partial sums.
   3. The limit of the original series.
   4. The limit of the derivatives.