Question

Identify the type of sequence (convergent or divergent):

∑ from n = 1 to [infinity] of (n / (n+1))

Identify the type of sequence (convergent or divergent):
∑ from n = 1 to [infinity] of (1 / (1+4n²))

Identify the type of sequence (convergent or divergent):
∑ from n = 1 to [infinity] of (1 / √(n+5))

Identify the type of sequence (convergent or divergent):
∑ from n = 1 to [infinity] of ?

Answer 1

The first series diverges, the second series converges, and the third series diverges. The nature of the last series cannot be determined without further information.

Step-by-step explanation:

To determine whether each series is convergent or divergent, we often use comparison tests or other convergence tests.

• For the series ∑ from n = 1 to ∞ of (n / (n+1)), this series diverges. As n approaches infinity, each term approaches 1, and summing infinitely many 1's leads to infinity.
• The series ∑ from n = 1 to ∞ of (1 / (1+4n²)) is convergent. By comparing it to the sum of 1/n², which is a p-series with p > 1, we can determine that it converges by the p-series test or comparison test.
• For the series ∑ from n = 1 to ∞ of (1 / √(n+5)), it diverges. This series behaves similarly to the harmonic series 1/n in the long run, and since the harmonic series diverges, so does this one.

It is not clear what the last series is, as the question seems to end with a question mark (?). Without the proper mathematical expression for the last series, it's impossible to determine its convergence.