How Telescoping Series Relate to Other Series Convergence Tests

📚 Quick Study Guide

• 🔍 Telescoping Series: A series where most terms cancel out, leaving only a few. Often takes the form $\sum_{n=1}^{\infty} (b_n - b_{n+1})$. Convergence is determined by the limit of the remaining terms.
• 💡 Convergence Condition: A telescoping series $\sum_{n=1}^{\infty} (b_n - b_{n+1})$ converges if and only if $\lim_{n \to \infty} b_n$ exists. If the limit exists (let's say it's $L$), then the sum of the series is $b_1 - L$.
• 📝 Divergence Condition: If $\lim_{n \to \infty} b_n$ does not exist, the telescoping series diverges.
• ➕ Relationship to Limit Test: The limit test states that if $\lim_{n \to \infty} a_n \neq 0$, then $\sum_{n=1}^{\infty} a_n$ diverges. Telescoping series can sometimes be simplified to relate back to this test. If the remaining terms after cancellation don't approach zero, the series diverges.
• ➖ Relationship to Integral Test (Indirect): While not directly related, the behavior of telescoping series (terms canceling out) can provide intuition for understanding how integrals approximate sums in the integral test. Both involve understanding how individual components contribute to an overall value.
• ➗ Relationship to Comparison Tests (Indirect): Telescoping series' convergence can sometimes be proven using comparison tests after algebraic manipulation. This involves finding a series whose convergence is already known and comparing the telescoping series to it.
• 📈 Partial Sums: The key to analyzing telescoping series is examining their partial sums. The $N^{th}$ partial sum, $S_N$, often simplifies dramatically, allowing us to easily find $\lim_{N \to \infty} S_N$.

Practice Quiz

1. What is a key characteristic of a telescoping series?
   1. A. All terms are positive.
   2. B. Most terms cancel out.
   3. C. The terms increase exponentially.
   4. D. The series is always divergent.
2. For a telescoping series $\sum_{n=1}^{\infty} (b_n - b_{n+1})$, what condition determines its convergence?
   1. A. $\lim_{n \to \infty} b_n = 0$
   2. B. $\lim_{n \to \infty} b_n$ exists.
   3. C. $\lim_{n \to \infty} b_n = \infty$
   4. D. $b_n$ is always positive.
3. If $\lim_{n \to \infty} b_n = L$ for the telescoping series $\sum_{n=1}^{\infty} (b_n - b_{n+1})$, what is the sum of the series?
   1. A. $L$
   2. B. $b_1$
   3. C. $b_1 - L$
   4. D. $L - b_1$
4. How does the Limit Test relate to the divergence of some telescoping series?
   1. A. It directly proves convergence.
   2. B. If simplified terms don't approach zero, the series diverges.
   3. C. It always guarantees convergence.
   4. D. It is not related to telescoping series.
5. What is the significance of partial sums when analyzing telescoping series?
   1. A. They always diverge.
   2. B. They often simplify dramatically.
   3. C. They are irrelevant.
   4. D. They always converge to zero.
6. Which of the following series is a telescoping series?
   1. A. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   2. B. $\sum_{n=1}^{\infty} \frac{1}{n}$
   3. C. $\sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1})$
   4. D. $\sum_{n=1}^{\infty} 2^n$
7. Suppose the partial sum of a telescoping series is $S_n = 3 - \frac{1}{n}$. What does the series converge to?
   1. A. 0
   2. B. 1
   3. C. 2
   4. D. 3