Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.)
For each statement, enter C (for "correct") if the argument is valid, or enter I (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.)
1. For all n > 2, ln(n)/n > 1n, and the series ∑1/n diverges, so by the Comparison Test, the series ∑ln/(n)n diverges.
2. For all n > 1, arctan(n)/n³ < π2n³, and the series π/2 ∑1/n³ converges, so by the Comparison Test, the series ∑arctan(n)/n³ converges.
3. For all n > 1, n/(2−n³) < 1n², and the series ∑1/n² converges, so by the Comparison Test, the series ∑n/(2−n³) converges.
4. For all n > 1, ln(n)/n² < 1/n¹.⁵, and the series ∑1/n^¹.⁵ converges, so by the Comparison Test, the series ∑ln(n)/n² converges.
5. For all n > 1, 1/nln(n) < 2/n, and the series 2 ∑1/n diverges, so by the Comparison Test, the series ∑1/nln(n) diverges.
6. For all n > 2, 1/(n²−7) < 1/n², and the series ∑1/n² converges, so by the Comparison Test, the series ∑1/(n²−7) converges.