Question

For which of the following series would the divergence test be appropriate to determine its convergence or divergence?

A. ∑ (n=1 to [infinity]) 5n⁴/n
B. ∑ (n=1 to [infinity]) 3n³ + 5n² + 6n + 4 / (5 + 6cos²(4n))
C. ∑ (n=1 to [infinity]) 8n³n + 4n³
D. None of the above

Answer 1

The divergence test is appropriate for determining the convergence or divergence of a series. It is appropriate for all the given series.

Step-by-step explanation:

The divergence test is appropriate for determining the convergence or divergence of a series when the terms of the series do not approach zero as n approaches infinity. It states that if the limit of the terms of the series as n approaches infinity does not equal 0, then the series diverges.

Using the divergence test, we can analyze the given series:

A. ∑ (n=1 to [infinity]) 5n⁴/n
Since the term 5n⁴/n does not approach 0 as n approaches infinity, we can use the divergence test.

B. ∑ (n=1 to [infinity]) 3n³ + 5n² + 6n + 4 / (5 + 6cos²(4n))
The expression inside the series involves trigonometric functions which are oscillatory and do not approach zero. Hence, we can use the divergence test.

C. ∑ (n=1 to [infinity]) 8n³n + 4n³
The term 8n³n + 4n³ does not approach zero as n approaches infinity, so the divergence test can be used.

D. None of the above
Since all the given series can be analyzed using the divergence test, the answer is None of the above.