Let #an and #bn be series with positive terms. Is each of the following statements true or false? If the statement is false, give an example that disproves the statement. (a) If #an and #bn are divergent, then #an bn is divergent. (b) If #an converges and #bn diverges, then #an bn diverges. (c) If #an and #bn are convergent, then #an bn is convergent
The Correct Answer and Explanation is:
ChatGPT said:
Correct Answers:
(a) False
(b) False
(c) True
Explanation (300 words, no use of the word we):
Consider each statement involving infinite series with positive terms:
(a) If ∑an and ∑bn are divergent, then ∑an·bn is divergent.
This is false. A counterexample proves the claim incorrect. Let an = 1/n and bn = 1/n. Both ∑(1/n) diverge (harmonic series). However, the product series an·bn = (1/n)·(1/n) = 1/n². The series ∑(1/n²) converges. This contradicts the statement. Therefore, divergence of two series does not guarantee divergence of their term-by-term product.
(b) If ∑an converges and ∑bn diverges, then ∑an·bn diverges.
This is also false. An example disproves the statement. Let an = 1/n² (convergent p-series with p = 2) and bn = 1 (constant sequence). Then an·bn = 1/n², and ∑(1/n²) converges. Here, ∑an·bn converges even though ∑bn diverges (since ∑1 diverges). Therefore, convergence of one series and divergence of the other does not necessarily imply divergence of their product.
(c) If ∑an and ∑bn are convergent, then ∑an·bn is convergent.
This is true for series with positive terms. When an ≥ 0 and bn ≥ 0, and both ∑an and ∑bn converge, the term-by-term product an·bn tends toward 0. Since an and bn are bounded and positive, the product an·bn is also bounded and positive. The comparison test can be used to compare the product series to another known convergent series. For example, since an is convergent, it is bounded by some M > 0. Then an·bn ≤ M·bn, and since ∑bn converges, the comparison test confirms convergence of ∑an·bn.
In conclusion: Statement (a) is false, proven by the harmonic series; statement (b) is false, shown using convergent p-series; statement (c) is true under the condition of positive terms.
