Question

Each of the following statements is an attempt to show that a given series is convergent or divergent not 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/n1.5 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.

Answer 1

1) The argument is not valid because if the bigger series diverges, the smaller series may converge or diverge.

Therefore, Argument is flawed.

(2) For all n >2, diverges. If the smaller series diverges, the bigger series also diverges.

Therefore, argument is correct.

(3) For all n > 1,

The series converges. Therefore, By comparison if the bigger series converges, smaller also converges.

Therefore, argument is correct.

(4) For all n >2,

Therefore, Argument is flawed.

(5) For all n >1,

and the series converges.

Therefore, By comparison test if bigger series converges, smaller also converges.

Explanation:

Please see the attached documents