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INTEGRAL TEST: CONVERGENCE and SUM ESTIMATION SOLUTIONS
2
3. Determine if the following series are convergent or divergent.
a.
Improper integral converges so the
series converges by integral test.
b.
Improper integral diverges so the
series diverges by integral test.
c.
Improper integral diverges so the
series diverges by integral test.
d.
Improper integral diverges so the
series diverges by integral test
e.
Improper integral converges so the
series converges by integral test.
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f.
Improper integral diverges so the
series diverges by integral test
g.
Improper integral converges so the
series converges by integral test.
h.
Improper integral diverges so the
series diverges by integral test
i.
Improper integral converges so the
series converges by integral test.
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j.
Improper integral diverges so the
series diverges by integral test
5
∞
5. Consider ∑
𝑛=1
1
𝑛3
a. Determine whether the series converges or diverges.
𝑏
∞
∞
1
1
1
1
1
1
∫ 3 𝑑𝑥 = lim ∫ 3 𝑑𝑥 = lim (− 2 + ) =
→ ∑ 3 is convergent .
𝑏→∞
𝑏→∞
𝑥
𝑥
2𝑏
2
2
𝑛
1
𝑛=1
1
b. Find the upper bound.
∞
∞
∞
1
1
1
1
3
∑ 3 = 1 + ∑ 3 ≤ 1 + ∫ 3 𝑑𝑥 = 1 + =
𝑛
𝑛
𝑥
2
2
𝑛=1
𝑛=2
∞
→
∑
𝑛=1
1
1
3
≤
3
𝑛
2
c. Approximate the sum by the partial sum, 𝑠5 and determine accuracy (estimate 𝑅𝑛 )
𝑠 ≈ 𝑠𝑛 = 1 +
1
1
1
1
+
+
+
= 1.18566
8
27
64
125
𝑏
error: 𝑅𝑛 ≤ lim ∫
𝑏→∞
5
1
1
𝑑𝑥 =
= 0.02
3
𝑥
50
d. How many terms are required to ensure that the approximate sum is accurate to within 0.001?
∞
𝑅𝑛 ≤ A, so you have to find n that will give that accuracy :
∫ 𝑓(𝑥)𝑑𝑥 ≤ 𝐴
𝑛
𝑏
𝑅𝑛 ≤ lim ∫
𝑏→∞
𝑛
1
1
𝑑𝑥 =
3
𝑥
2𝑛2
1
≤ 0.001
2𝑛2
∞
e. Estimate the sum ∑
𝑛=1
𝑛2 ≥ 500 → 𝑛 ≥ 22.4 → 23 terms
1
with n = 5
𝑛3
∞
∞
𝑠𝑛 + ∫ 𝑓(𝑥)𝑑𝑥 ≤ 𝑠 ≤ 𝑠𝑛 + ∫ 𝑓(𝑥)𝑑𝑥
𝑛+1
𝑛
∞
𝑠5 = 1.18566
1
1
∫ 3 𝑑𝑥 =
= 0.02
𝑥
50
5
1.19955 < 𝑠 < 1.20566
∞
∫
6
1
1
𝑑𝑥 =
= 0.01389
𝑥3
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