1. Alternating Series Test
Section 9.5
Calculus BC AP/Dual, Revised ©2018
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§9.5: Alternating Series Test

2. Summary of Tests for Series
Looking at the first few terms of the sequence of partial sums may not help us much so we will learn the following ten tests to determine convergence or divergence:
P 𝒑-series: Is the series in the form 𝟏 𝒏 𝑷 ?
A Alternating series: Does the series alternate? If it does, are the terms getting smaller, and is the 𝒏th term 0?
R Ratio Test: Does the series contain things that grow very large as 𝒏 increases (exponentials or factorials)?
R Root Test: Does the series contain a radical?
T Telescoping series: Will all but a couple of the terms in the series cancel out?
I Integral Test: Can you easily integrate the expression that define the series?
N 𝒏th Term divergence Test: Is the nth term something other than zero?
G Geometric series: Is the series of the form, 𝒏=𝟎 ∞ 𝒂 𝒓 𝒏
C Comparison Tests: Is the series almost another kind of series (e.g. 𝒑-series or geometric)? Which would be better to use: Direct or Limit Comparison Test?
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§9.5: Alternating Series Test

3. Alternating Series Test
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§9.5: Alternating Series Test

4. Alternating Series Test Sum
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§9.5: Alternating Series Test

5. Alternating Series Test
An alternating series is a series whose terms are alternatively positive and negative on consecutive terms
If 𝒂 𝒏 >𝟎, then the alternating series 𝒏=𝟏 ∞ −𝟏 𝒏 𝒂 𝒏 and 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝒂 𝒏 converges if the following conditions are met:
Alternates in signs
Decreases in magnitude 𝒂 𝒏+𝟏 < 𝒂 𝒏 for all 𝒏
Have a limit of zero 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 =𝟎
This does not say if 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎, the series diverges by the AST. The AST CAN only be used to prove convergence. If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎, then the series diverges by the nth term test for divergence and not by the AST.
Alternate Harmonic Series is NOT like the harmonic series. It will allow the series to converge to the sum of zero.
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§9.5: Alternating Series Test

6. Alternating Series Test Examples

7. §9.5: Alternating Series Test
Example 1
Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝒏 .
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§9.5: Alternating Series Test

8. §9.5: Alternating Series Test
Example 1
Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝒏 .
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§9.5: Alternating Series Test

9. §9.5: Alternating Series Test
Example 1 (other way)
Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝒏 .
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§9.5: Alternating Series Test

10. §9.5: Alternating Series Test
Example 2
Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝐥𝐧 𝒏+𝟏 .
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§9.5: Alternating Series Test

11. §9.5: Alternating Series Test
Example 3
Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝒏 𝟐 𝒏 𝟐 +𝟓 .
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§9.5: Alternating Series Test

12. §9.5: Alternating Series Test
Example 4
Prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝐜𝐨𝐬 𝒏𝝅 𝒏 .
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§9.5: Alternating Series Test

13. §9.5: Alternating Series Test
Your Turn
Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏−𝟏 𝒏! . (Use the comparison for decreasing)
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§9.5: Alternating Series Test

14. Absolute vs Conditional Convergent
Absolute Convergence is where the series of 𝒏=𝟏 ∞ 𝒂 𝒏 converges, then 𝒏=𝟏 ∞ 𝒂 𝒏 also converges.
Conditionally Convergence is where the series of 𝒏=𝟏 ∞ 𝒂 𝒏 converges, then 𝒏=𝟏 ∞ 𝒂 𝒏 diverges
Sometimes, a rearrangement of terms in a convergent alternating series can result in a different sum.
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§9.5: Alternating Series Test

15. §9.5: Alternating Series Test
Example 5
Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent.
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§9.5: Alternating Series Test

16. §9.5: Alternating Series Test
Example 5
Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent.
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§9.5: Alternating Series Test

17. §9.5: Alternating Series Test
Example 6
Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟏 𝟑 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent.
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§9.5: Alternating Series Test

18. §9.5: Alternating Series Test
Example 6
Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟏 𝟑 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent.
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§9.5: Alternating Series Test

19. §9.5: Alternating Series Test
Your Turn
Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟐 𝒏 𝟐 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent.
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§9.5: Alternating Series Test

20. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following series converge? I. 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 II. 𝒏=𝟏 ∞ 𝟏 𝒏 III. 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 (A) I only (B) II only (C) III only (D) I and III only
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§9.5: Alternating Series Test

21. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following series converge?
Vocabulary
Connections and Process
Answer
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§9.5: Alternating Series Test

22. §9.5: Alternating Series Test
Assignment
Page EOO, 27, 29, odd
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§9.5: Alternating Series Test