Question

Determine for each series whether it converges or diverges. Justify your answers.

aₙ = √ (n+1/4n+7)

Answer 1

To determine whether the series
`\( \sum_(n=1)^(\infty) \sqrt{(n+1)/(4n+7)} \)` converges or diverges, we can use the comparison test or limit comparison test. The series
`\( \sum_(n=1)^(\infty) (1)/(√(n)) \)` is a well-known convergent p-series with p =
`(1)/(2) \)`, and it converges. Therefore, by the limit comparison test,
`\( \sum_(n=1)^(\infty) \sqrt{(n+1)/(4n+7)} \)` also converges.

Step-by-step explanation:

To determine whether the series
`\( \sum_(n=1)^(\infty) \sqrt{(n+1)/(4n+7)} \)` converges or diverges, we can use the comparison test or limit comparison test.

Let's consider the limit comparison test. We'll compare the given series to a known series whose convergence behavior is well-known. We can choose the series
`\( \sum_(n=1)^(\infty) (1)/(√(n)) \)` .

1. First, observe that
`\( (n+1)/(4n+7) \)` is asymptotically similar to
`\( (1)/(4) \)` as n becomes large.

2. So, consider the series
`\( \sum_(n=1)^(\infty) \sqrt{(1)/(4) \cdot (n+1)/(n)} \)` .

3. Simplify the expression inside the square root:
`\( \sqrt{(1)/(4) \cdot (n+1)/(n)} = \sqrt{(n+1)/(4n)} = (1)/(2) \sqrt{(n+1)/(n)} \)`.

4. Now, compare this series to
`\( \sum_(n=1)^(\infty) (1)/(√(n)) \)`. If
`\( \sum_(n=1)^(\infty) (1)/(√(n)) \)` converges, then
`\( \sum_(n=1)^(\infty) \sqrt{(n+1)/(4n+7)} \)` converges.

The series
`\( \sum_(n=1)^(\infty) (1)/(√(n)) \)` is a well-known convergent p-series with p =
`(1)/(2) \)`, and it converges.

Therefore, by the limit comparison test,
`\( \sum_(n=1)^(\infty) \sqrt{(n+1)/(4n+7)} \)` also converges.