Let \(\rm \Sigma_{n=1}^{\infty}a_n\) be a convergent series of real numbers. For n ≥ 1 define
\(\rm A_n=\left\{\begin{matrix}a_n, &if\ a_n >0\\\ 0, &otherwise\end{matrix}\right., \rm B_n=\left\{\begin{matrix}a_n, &if\ a_n <0\\\ 0, &otherwise\end{matrix}\right.\)
Which of the following statements are necessarily true?
1
An → 0 and Bn → 0 as n → ∞
2
If \(\rm \Sigma_{n=1}^{\infty}a_n\) is absolutely convergent, then both \(\rm \Sigma_{n=1}^{\infty}A_n\ and \Sigma_{n=1}^{\infty}B_n\) are absolutely convergent
3
Both \(\rm \Sigma_{n=1}^{\infty}A_n\ and \Sigma_{n=1}^{\infty}B_n\) are convergent.
4
If \(\rm \Sigma_{n=1}^{\infty}a_n\) is not absolutely convergent, then both \(\rm \Sigma_{n=1}^{\infty}A_n\ and \Sigma_{n=1}^{\infty}B_n\) are divergent