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How do you do these 2 questions?

How do you do these 2 questions?-example-1
How do you do these 2 questions?-example-1
How do you do these 2 questions?-example-2

1 Answer

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Answer:

0 ≤ an ≤ bn

The series ∑₁°° bn converges

The series ∑₁°° an converges by comparison to ∑₁°° bn.

0 ≤ an ≤ bn

The series ∑₁°° bn diverges

The comparison test is inconclusive for our choice of bn.

Explanation:

an = 1 / (n² + n + 3) and bn = 1 / n²

The numerators are the same, and the denominator of an is greater than the denominator of bn. So 0 ≤ an ≤ bn.

bn is a p series with p > 1, so it converges.

Since the larger function converges, the smaller function also converges.

an = (3n − 1) / (6n² + 2n + 1) and bn = 1 / (2n)

If we rewrite bn as bn = (3n − 1) / (6n² − 2n), we can tell that when the numerators are equal, the denominator of an is greater than the denominator of bn. So 0 ≤ an ≤ bn.

bn is a p series with p = 1, so it diverges.

The larger function diverges. We cannot conclude whether the smaller function converges or diverges.

User Richard Kamere
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