AP Calculus BC 10.9 Determining Absolute or Conditional Convergence Study Notes - New Syllabus
AP Calculus BC 10.9 Determining Absolute or Conditional Convergence Study Notes- New syllabus
AP Calculus BC 10.9 Determining Absolute or Conditional Convergence Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Applying limits may allow us to determine the finite sum of infinitely many terms.
Key Concepts:
- Determining Absolute or Conditional Convergence
Determining Absolute or Conditional Convergence
Determining Absolute or Conditional Convergence
When a series converges, it can do so in two distinct ways absolutely or conditionally. Understanding the difference is crucial, especially for series with both positive and negative terms.
Absolute Convergence:
A series \( \displaystyle \sum a_n \) is said to converge absolutely if the series formed by taking the absolute values of its terms, \( \displaystyle \sum |a_n| \), converges.
If a series converges absolutely, then it also converges (this is always true).
Conditional Convergence:
A series \( \displaystyle \sum a_n \) is said to converge conditionally if \( \displaystyle \sum a_n \) converges, but \( \displaystyle \sum |a_n| \) diverges.
Conditional convergence means that the cancellation between positive and negative terms is essential for convergence without it (taking absolute values), the series would diverge.
Steps to Determine Absolute or Conditional Convergence:
- First, test \( \displaystyle \sum |a_n| \) using appropriate convergence tests (Ratio, Root, Comparison, p-Series, etc.).
- If \( \displaystyle \sum |a_n| \) converges → the original series converges absolutely.
- If \( \displaystyle \sum |a_n| \) diverges, check the original \( \displaystyle \sum a_n \) for convergence using tests for alternating series (e.g., Alternating Series Test) or other applicable methods.
- If the original series converges but the absolute series diverges → conditional convergence.
- If the original series diverges → neither absolute nor conditional convergence applies (simply divergent).
When this topic is most useful:
- Alternating series where terms decay slowly (like \( \frac{1}{n} \), \( \frac{\sin n}{n} \))
- Series with factorials or exponentials but with alternating signs
- Fourier-type expansions and oscillatory sums
Example
Determine whether \( \displaystyle \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^2} \) is absolutely or conditionally convergent.
▶️ Answer/Explanation
Test absolute convergence:
\( \displaystyle \sum |a_n| = \sum_{n=1}^\infty \dfrac{1}{n^2} \) is a p-series with \( p = 2 > 1 \), so it converges.
Conclusion: Since the absolute value series converges, the original series converges absolutely.
Final Answer: Absolutely convergent.
Example
Determine whether \( \displaystyle \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n} \) is absolutely or conditionally convergent.
▶️ Answer/Explanation
Test absolute convergence:
\( \displaystyle \sum |a_n| = \sum_{n=1}^\infty \dfrac{1}{n} \) is the harmonic series, which diverges.
Test original series for convergence:
This is an alternating harmonic series: terms decrease in magnitude to 0 and alternate in sign. By the Alternating Series Test, it converges.
Conclusion: Since the absolute series diverges but the original converges, it is conditionally convergent.
Final Answer: Conditionally convergent.
Example
For the series \( \displaystyle \sum_{n=1}^\infty \dfrac{(-1)^n}{\sqrt{n}} \):
A. Absolutely convergent
B. Conditionally convergent
C. Divergent
D. Test is inconclusive
▶️ Answer/Explanation
Test absolute convergence:
\( \displaystyle \sum |a_n| = \sum_{n=1}^\infty \dfrac{1}{\sqrt{n}} \) is a p-series with \( p = \dfrac12 < 1 \), so it diverges.
Test original series for convergence:
It is an alternating series with terms decreasing in magnitude to 0, so by the Alternating Series Test, it converges.
Conclusion: Divergent in absolute value but convergent in original form → conditionally convergent.
Final Answer: B