Calculus 10.

1 Convergent and Divergent Infinite Series Notes

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Recall: Writing terms of a sequence.
𝑎𝑎𝑛𝑛 = {1 + (−2)𝑛𝑛 }

−1, 5, −7, 17, −31

Sequence: A collection of numbers that are in one-to-one correspondence with positive

integers.
−2 4 26 80 242
− −
6 24 120
𝑛𝑛 = 1 2 3 4 5

Monotonic Sequences Bounded Sequences

never decreases or never increases
𝑎𝑎1 ≤ 𝑎𝑎2 ≤ 𝑎𝑎3 ≤ ⋯ ≤ 𝑎𝑎𝑛𝑛 𝑎𝑎𝑛𝑛 ≤ 𝑀𝑀 (upper bound / above)
or 𝑎𝑎𝑛𝑛 ≥ 𝑁𝑁 (lower bound / below)
𝑎𝑎1 ≥ 𝑎𝑎2 ≥ 𝑎𝑎3 ≥ ⋯ ≥ 𝑎𝑎𝑛𝑛 {𝑎𝑎𝑛𝑛 } bounded if both are true

Infinite Series:
∞

� 𝑎𝑎𝑛𝑛 = 𝑎𝑎1 + 𝑎𝑎2 + 𝑎𝑎3 + ⋯ + 𝑎𝑎𝑛𝑛

𝑛𝑛=1

Partial Sum:
𝑆𝑆𝑛𝑛 = 𝑎𝑎1 + 𝑎𝑎2 + 𝑎𝑎3 + ⋯ + 𝑎𝑎𝑛𝑛

𝒂𝒂𝒏𝒏 vs 𝑺𝑺𝒏𝒏 :

𝑎𝑎𝑛𝑛 is an expression that gives the nth term in a sequence.

𝑆𝑆𝑛𝑛 is an expression that gives the sum of the first n terms.

1. Use the following sequence 2, 4, 6, 8, 10 to find 𝑎𝑎4 and 𝑆𝑆4 .

� 𝑎𝑎𝑛𝑛 = lim 𝑆𝑆𝑛𝑛

𝑛𝑛→∞
𝑛𝑛=1
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Convergent and Divergent Series
∞

For the infinite series � 𝑎𝑎𝑛𝑛 , the nth partial sum is 𝑆𝑆𝑛𝑛 = 𝑎𝑎1 + 𝑎𝑎2 + 𝑎𝑎3 + ⋯ + 𝑎𝑎𝑛𝑛 .
𝑛𝑛=1 ∞

If the sequence of the partial sum {𝑆𝑆𝑛𝑛 } converges to 𝑆𝑆, then the series � 𝑎𝑎𝑛𝑛 converges. The
limit 𝑆𝑆 is called the sum of the series. 𝑛𝑛=1

Likewise, if {𝑆𝑆𝑛𝑛 } diverges, then the series diverges.

∞
2. Does the series converge or diverge? � 1
2𝑛𝑛
𝑛𝑛=1

∞
10
3. Use a calculator to find the partial sum 𝑆𝑆𝑛𝑛 of the series � for 𝑛𝑛 = 200, 1000.
𝑛𝑛(𝑛𝑛 + 2)
𝑛𝑛=1

∞
4. Does the series converge or diverge? � 𝑛𝑛
𝑛𝑛=1

10.1 Convergent and Divergent Infinite Series

Calculus
Practice
∞

1. Given the infinite series �(−1)𝑛𝑛, find the sequence of partial sums 𝑆𝑆1 , 𝑆𝑆2 , 𝑆𝑆3 , 𝑆𝑆4 , and 𝑆𝑆5.
𝑛𝑛=1
 1 1 1 1 1
2. Find the sequence of partial sums 𝑆𝑆1 , 𝑆𝑆2 , 𝑆𝑆3 , 𝑆𝑆4 , and 𝑆𝑆5 for the infinite series 1 + + + + + + ⋯.
2 4 6 8 10

3. If the infinite series � 𝑎𝑎𝑛𝑛 has 𝑛𝑛th partial sum 𝑆𝑆𝑛𝑛 = (−1)𝑛𝑛+1 for 𝑛𝑛 ≥ 1, what is the sum of the series?
𝑛𝑛=1

∞
𝑛𝑛
4. The infinite series � 𝑎𝑎𝑛𝑛 has 𝑛𝑛th partial sum 𝑆𝑆𝑛𝑛 = for 𝑛𝑛 ≥ 1. What is the sum of the series?
4𝑛𝑛+1
𝑛𝑛=1

∞
6
5. Use a calculator to find the partial sum 𝑆𝑆𝑛𝑛 of the series � for 𝑛𝑛 = 100, 500, 1000.
𝑛𝑛(𝑛𝑛 + 3)
𝑛𝑛=1

6. Show that the sequence with the given 𝑛𝑛th term 𝑎𝑎𝑛𝑛 = 1 + 2𝑛𝑛 is monotonic.

∞
1
7. What is the 𝑛𝑛th partial sum of the infinite series � ?
2𝑛𝑛+1
𝑛𝑛=1
10.1 Convergent and Divergent Infinite Series Test Prep
∞
1
8. Which of the following could be the 𝑛𝑛th partial sum for the infinite series � ?
4𝑛𝑛
𝑛𝑛=1

1 1 1 1 1 1 1 1
(A) 𝑆𝑆𝑛𝑛 = �1 + � (B) 𝑆𝑆𝑛𝑛 = �1 − � (C) 𝑆𝑆𝑛𝑛 = �1 − � (D) 𝑆𝑆𝑛𝑛 = �1 − �
3 4 𝑛𝑛 3 4 𝑛𝑛+1 3 4 𝑛𝑛 4 3𝑛𝑛

∞
7
9. If the infinite series � 𝑎𝑎𝑛𝑛 is convergent and has a sum of , which of the following could be the 𝑛𝑛th partial
8
sum? 𝑛𝑛=1

7𝑛𝑛+1 7𝑛𝑛2 +1
(A) 𝑆𝑆𝑛𝑛 = (B) 𝑆𝑆𝑛𝑛 =
8𝑛𝑛2 +1 8𝑛𝑛+1

7 1 1 7 1 1
(C) 𝑆𝑆𝑛𝑛 = 2 � − − � (D) 𝑆𝑆𝑛𝑛 = � − − �
8 𝑛𝑛+2 𝑛𝑛+3 8 𝑛𝑛+2 𝑛𝑛+3

10. Which of the following sequences with the given 𝑛𝑛th term is bounded and monotonic?

𝑛𝑛2 3𝑛𝑛 cos 𝑛𝑛

(A) 𝑎𝑎𝑛𝑛 = 2 + (−1)𝑛𝑛 (B) 𝑎𝑎𝑛𝑛 = (C) 𝑎𝑎𝑛𝑛 = (D) 𝑎𝑎𝑛𝑛 = 𝑛𝑛
𝑛𝑛+1 𝑛𝑛+2