AP Calculus BC 10.3 The nth Term Test for Divergence Study Notes - New Syllabus
AP Calculus BC 10.3 The nth Term Test for Divergence Study Notes- New syllabus
AP Calculus BC 10.3 The nth Term Test for Divergence 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:
- The \( n \)th Term Test for Divergence
The \( n \)th Term Test for Divergence
The \( n \)th Term Test for Divergence
The \( n \)th Term Test for Divergence is used to check whether a series definitely diverges. It is based on the behavior of the terms \( a_n \) as \( n \to \infty \).
The Test:
- If \( \displaystyle \lim_{n \to \infty} a_n \neq 0 \) or the limit does not exist, then the series \( \displaystyle \sum a_n \) diverges.
- If \( \displaystyle \lim_{n \to \infty} a_n = 0 \), the test is inconclusive the series may converge or diverge, and other tests must be used.
Important Note:
- For a series to converge, the terms must get smaller and approach zero. If they do not approach zero, the sum will either grow without bound or oscillate, making convergence impossible.
- This test is often called the Test for Divergence rather than the Test for Convergence because it can only prove divergence, never convergence.
- Even if \( \lim_{n \to \infty} a_n = 0 \), the series can still diverge for example, the harmonic series \( \displaystyle \sum_{n=1}^\infty \dfrac{1}{n} \) has terms approaching zero but still diverges.
- If \( \lim_{n \to \infty} a_n \) does not exist (terms fail to settle on a single value), the series automatically diverges.
- The \(n\)th term test should always be applied first when analyzing a series because it can quickly identify divergence before trying more complex tests.
Reasoning:
If the sum \( S = a_1 + a_2 + a_3 + \dots \) converges to a finite value, the terms being added must get smaller and approach zero. If they fail to approach zero, the sum will grow indefinitely or oscillate without settling.
Example
Test the series \( 5 + 5 + 5 + 5 + \dots \) for divergence using the \( n \)th Term Test.
▶️ Answer/Explanation
Here, \( a_n = 5 \).
\( \displaystyle \lim_{n \to \infty} a_n = \lim_{n \to \infty} 5 = 5 \neq 0 \).
Since the limit is not zero, the series diverges by the \( n \)th Term Test.
Final Answer: Divergent.
Example
Test the series \( \dfrac{1}{n} \) for divergence using the \( n \)th Term Test.
▶️ Answer/Explanation
Here, \( a_n = \dfrac{1}{n} \).
\( \displaystyle \lim_{n \to \infty} a_n = \lim_{n \to \infty} \dfrac{1}{n} = 0 \).
Since the limit is zero, the test is inconclusive. The series may converge or diverge, so another test (like the Harmonic Series Test) is needed.
In fact, \( \displaystyle \sum_{n=1}^\infty \dfrac{1}{n} \) is the harmonic series, which is known to diverge.
Final Answer: Limit is zero — inconclusive by this test, but the series diverges by other means.
Example
Test the series \( \dfrac{3n+1}{n} \) for divergence using the \( n \)th Term Test.
▶️ Answer/Explanation
Here, \( a_n = \dfrac{3n+1}{n} = 3 + \dfrac{1}{n} \).
\( \displaystyle \lim_{n \to \infty} a_n = 3 + 0 = 3 \neq 0 \).
Since the limit is not zero, the series diverges by the \( n \)th Term Test.
Final Answer: Divergent.