1. Nth, Geometric, and Telescoping Test
Section 9.2
Calculus BC AP/Dual, Revised ©2018
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§9.2: nth, Geometric, and Telescoping 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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§9.2: nth, Geometric, and Telescoping Test

3. §9.2: nth, Geometric, and Telescoping Test
Definitions
Series is the sum of the terms in a sequence.
Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
A series can be written with summation symbol sigma, , the Greek letter “𝑺”.
𝑺𝒏 is often called an 𝒏𝒕𝒉 partial sum, since it can represent the sum of a certain ‘part’ of a sequence.
Infinite series converges if the sequence of partial sums converges to some number, 𝑺𝒏.
If the sequence of partial sums diverges, then the series diverges.
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§9.2: nth, Geometric, and Telescoping Test

4. What is Sigma Notation? "The summation from 𝟏 to ∞ of 𝟐𝒏+𝟏":
Upper Limit
Function
Summation
Lower Limit
Known as “index”
"The summation from 𝟏 to ∞ of 𝟐𝒏+𝟏":
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§9.2: nth, Geometric, and Telescoping Test

5. §9.2: nth, Geometric, and Telescoping Test
Example 1
Given the series 𝒏=𝟏 ∞ 𝒏 𝟐 +𝟐 , find the first five terms of the sequence of partial sums, and list them below. Then, evaluate. Is the sequence of partial sums has a limit or bound?
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§9.2: nth, Geometric, and Telescoping Test

6. §9.2: nth, Geometric, and Telescoping Test
Geometric Series Test
Geometric series is in the form, 𝒏=𝟏 ∞ 𝒂 𝟏 ( 𝒓) 𝒏 or 𝒏=𝟎 ∞ 𝒂 𝟏 ( 𝒓) 𝒏−𝟏 ;𝒂≠𝟎
𝒂 𝟏 is the Initial term of the series
𝒓 is the common ratio
Convergence vs. Divergence
The geometric series converges if 𝒓 <𝟏 to the sum of 𝑺= 𝒂 𝟏 𝟏−𝒓
The geometric series diverges if 𝒓 ≥𝟏
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§9.2: nth, Geometric, and Telescoping Test

7. §9.2: nth, Geometric, and Telescoping Test
Example 2
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝟐 𝒏 If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

8. §9.2: nth, Geometric, and Telescoping Test
Example 2
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝟐 𝒏 If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

9. §9.2: nth, Geometric, and Telescoping Test
Example 3
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ − 𝟑 𝟐 𝒏
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§9.2: nth, Geometric, and Telescoping Test

10. §9.2: nth, Geometric, and Telescoping Test
Example 4
Determine whether the following series converge or diverge, 𝒏=𝟎 ∞ 𝟑 𝒏+𝟏 𝟓 𝒏
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§9.2: nth, Geometric, and Telescoping Test

11. §9.2: nth, Geometric, and Telescoping Test
Example 4
Determine whether the following series converge or diverge, 𝒏=𝟎 ∞ 𝟑 𝒏+𝟏 𝟓 𝒏
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§9.2: nth, Geometric, and Telescoping Test

12. §9.2: nth, Geometric, and Telescoping Test
Your Turn
Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟑 − 𝟏 𝟐 𝒏 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

13. Nth Term Test for Divergence
One Way Test (Divergence Only)
If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎 , then the series 𝒏=𝟏 ∞ 𝒂 𝒏 diverges
Therefore, if 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 =𝟎 , then the series DOES NOT converge
If the test does not pass, the test is INCONCLUSIVE and another test must be used
Use this test FIRST before others, due to time constraints
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§9.2: nth, Geometric, and Telescoping Test

14. §9.2: nth, Geometric, and Telescoping Test
Example 5
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟐𝒏+𝟑 𝟑𝒏−𝟓 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

15. §9.2: nth, Geometric, and Telescoping Test
Example 6
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝒏! 𝟐𝒏!+𝟏 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

16. §9.2: nth, Geometric, and Telescoping Test
Example 7
Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟏 𝟏.𝟏 𝒏 .
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§9.2: nth, Geometric, and Telescoping Test

17. §9.2: nth, Geometric, and Telescoping Test
Example 7
Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟏 𝟏.𝟏 𝒏 .
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§9.2: nth, Geometric, and Telescoping Test

18. §9.2: nth, Geometric, and Telescoping Test
Your Turn
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝒏 −𝟐 𝟑 𝒏 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

19. §9.2: nth, Geometric, and Telescoping Test
Known as the terms “collapse” to one term or several terms
Uses the associative property of addition
Series collapses to a finite sum
To get the sum, start plugging in numbers
Hint: Generally has two ratios associated with Telescoping Series when generating the sum
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§9.2: nth, Geometric, and Telescoping Test

20. §9.2: nth, Geometric, and Telescoping Test
Example 8
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝟐𝒏+𝟏 − 𝟏 𝟐𝒏+𝟑 𝒏=𝟏 ∞ 𝟏 𝟐𝒏+𝟏 − 𝟏 𝟐𝒏+𝟑 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

21. §9.2: nth, Geometric, and Telescoping Test
Example 9
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

22. §9.2: nth, Geometric, and Telescoping Test
Example 9
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

23. §9.2: nth, Geometric, and Telescoping Test
Example 9
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

24. §9.2: nth, Geometric, and Telescoping Test
Example 9
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

25. §9.2: nth, Geometric, and Telescoping Test
Your Turn
Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝒏+𝟏 If it converges, identify the sum.
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§9.2: nth, Geometric, and Telescoping Test

26. AP Multiple Choice Practice Question 1 (non-calculator)
What is the value of 𝒏=𝟏 ∞ 𝟐 𝒏+𝟏 𝟑 𝒏 ? (A) 𝟐 (B) 𝟒 (C) 𝟔 (D) The series diverges
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§9.2: nth, Geometric, and Telescoping Test

27. AP Multiple Choice Practice Question 1 (non-calculator)
What is the value of 𝒏=𝟏 ∞ 𝟐 𝒏+𝟏 𝟑 𝒏 ?
Vocabulary
Connections and Process
Answer
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§9.2: nth, Geometric, and Telescoping Test

28. §9.2: nth, Geometric, and Telescoping Test
Assignment
Page odd, odd, odd
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§9.2: nth, Geometric, and Telescoping Test