Question

Let \(\left\{a_n\right\}^{\infin}_{n=1}\) be a sequence of real numbers.
Then, which of the following statements is/are always TRUE ?

• A. If \(\sum\limits_{n=1}^{\infin}a_n\)converges absolutely, then \(\sum\limits_{n=1}^{\infin}a_n^2\) converges absolutely
• B. If \(\sum\limits_{n=1}^{\infin}a_n\)converges absolutely, then \(\sum\limits_{n=1}^{\infin}a_n^3\) converges absolutely
• C. If \(\sum\limits_{n=1}^{\infin}a_n\) converges, then \(\sum\limits_{n=1}^{\infin}a^2_n\) converges
• D. If \(\sum\limits_{n=1}^{\infin}a_n\) converges, then \(\sum\limits_{n=1}^{\infin}a^3_n\) converges

Answer 1

The Correct Option is A, B

(A) If the series \( \sum_{n=1}^{\infty} a_n \) converges absolutely, then \( \sum_{n=1}^{\infty} a_n^2 \) converges absolutely. This is true because if \( \sum_{n=1}^{\infty} |a_n| \) converges, then \( |a_n| \) tends to 0 as \( n \) increases. Since \( a_n^2 \) is smaller than or equal to \( |a_n| \) for all \( n \), it follows that \( \sum_{n=1}^{\infty} a_n^2 \) also converges absolutely.
(B) If \( \sum_{n=1}^{\infty} a_n \) converges absolutely, then \( \sum_{n=1}^{\infty} a_n^3 \) converges absolutely. This is also true because the cube of a small number tends to 0 even faster than the number itself. Therefore, if \( \sum_{n=1}^{\infty} |a_n| \) converges, so will \( \sum_{n=1}^{\infty} |a_n^3| \), and the series \( \sum_{n=1}^{\infty} a_n^3 \) converges absolutely.
(C) This statement is not always true. The convergence of \( \sum_{n=1}^{\infty} a_n \) does not guarantee that \( \sum_{n=1}^{\infty} a_n^2 \) will converge. For example, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) converges, but \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) does not.
(D) Similarly, the convergence of \( \sum_{n=1}^{\infty} a_n \) does not imply that \( \sum_{n=1}^{\infty} a_n^3 \) converges. For example, if \( a_n = \frac{1}{n} \), then \( \sum_{n=1}^{\infty} a_n \) converges, but \( \sum_{n=1}^{\infty} a_n^3 \) diverges.
Therefore, the correct answers are (A) and (B).