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Real Analysis Question on Sequences and Series

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 ∑n=1∞an\sum\limits_{n=1}^{\infin}a_nconverges absolutely, then ∑n=1∞an2\sum\limits_{n=1}^{\infin}a_n^2 converges absolutely

B

If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_nconverges absolutely, then ∑n=1∞an3\sum\limits_{n=1}^{\infin}a_n^3 converges absolutely

C

If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n converges, then ∑n=1∞an2\sum\limits_{n=1}^{\infin}a^2_n converges

D

If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n converges, then ∑n=1∞an3\sum\limits_{n=1}^{\infin}a^3_n converges

Answer

If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_nconverges absolutely, then ∑n=1∞an2\sum\limits_{n=1}^{\infin}a_n^2 converges absolutely

Explanation

Solution

The correct option is (A) : If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_nconverges absolutely, then ∑n=1∞an2\sum\limits_{n=1}^{\infin}a_n^2 converges absolutely and (B) : If ∑n=1∞an\sum\limits_{n=1}^{\infin}a_nconverges absolutely, then ∑n=1∞an3\sum\limits_{n=1}^{\infin}a_n^3 converges absolutely