$\sum a_{n}^{2}$ and $\sum b_{n}^{2}$ converges . Then what can be concluded about the convergence of $\sum a_{n}b_{n}$

Question

If two series converges and one of them is absolutely convergent then the product is absolutely convergent . Is it gives any conclusion here$\sum a_{n}^{2}$ and $\sum b_{n}^{2}$ converges . Then what can be concluded about the convergence of $\sum a_{n}b_{n}$

Answer 1

Since $\sum a_{n}^{2}$ and $\sum b_{n}^{2}$ are convergent, so is $\sum a_{n}^{2}+b_{n}^{2}$.

Then for each positive integer $n$, we have $$ |a_nb_n| = |a_n||b_n| \le \bigl(\max(|a_n|,|b_n|)\bigr)^2 \le a_{n}^{2}+b_{n}^{2} $$ hence $\sum |a_nb_n|$ is convergent.

It follows that $\sum a_nb_n$ is convergent.

Answer 2

Just use cauchy schwarz inequality: |(ambm+a(m+1)b(m+1)+..+anbn)|<=(am^2+..+an^2)^1/2 * (bm^2+..bn^2)^1/2 Right side can be sufficiently small using Cauchy criteria fot convergence of series an^2 &bn^2.

Answer 3

Since $0 \le (|a_n|-|b_n|)^2$, we have

$2|a_nb_n| \le a_n^2+b_n^2$.

Conclusion ?