Let $\{a_n\}$ be a sequence of real numbers. which one of the following is always true?

Question

Let $\{a_n\}$ be a sequence of real numbers. which one of the following is always true?

(a)$\sum a_n$ converges, then $\sum a_n^4$ converges.

(b)$\sum |a_n|$ converges, then $\sum a_n^2$ converges.

(c)$\sum a_n$ diverges, then $\sum a_n^3$ diverges.

(d)$\sum |a_n|$ diverges, then $\sum a_n^2$ diverges.

$\sum a_n$ converges $\implies lim_{n\rightarrow \infty}a_n=0$, But this won't help here. I know the fact that converse of the Ratio test, root test,...etc need not true.

(c)$a_n=\frac{1}{n}$ $\implies$ the statement is false.

(d)$a_n=\frac{(-1)^n}{n}$ $\implies$ the statement is false.

I am not able to judge (a) and (b). please help me.

Answer 1

(b) If $\sum |a_n|$) converges, then $a_n\to 0$ and hence $|a_n|\le 1$ for $n\gg 0$. Thus $0\le a_n^2\le |a_n|$ for almost all $n$. Conclude that $\sum a_n^2$ converges.

(a) Consider $a_n=\frac{(-1)^n}{\sqrt[4]n}$

Answer 2

For b), find some $N$ such that $n\geq N$, $|a_{n}|<1$, then $a_{n}^{2}\leq |a_{n}|$, so comparison test goes through. For a), consider $a_{n}=(-1)^{n}\dfrac{1}{n^{1/4}}$.