Which of the following series is not absolutely convergent?

Question

Which of the following series is not absolutely convergent?

Answer 1

For a series ∑a_n to be absolutely convergent ∑ l a_n l must be convergent.

Inserting absolute value symbols:

Choice 1 becomes ∑ (1/3)ⁿ, which is a convergent Geometric Series with r = 1/3

Choice 2 becomes ∑ 1/n^5/2, which is a convergent p-series with p = 5/2.

Choice 3 becomes ∑1/n^1/2, which is a divergent p-series with p = 1/2

Choice 4 becomes ∑ [1 / (n⁴+1)] which converges by comparing it to ∑1/n⁴ using the Comparison Test.