Help please!

Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.

I can prove that $\sum a_n b_n$ converges but couldn't for $\sum \sqrt{a_n b_n}$.

Thank you.

EDIT: $a_n, b_n \ge 0 \;\forall n$

  • 1
    $\begingroup$ I assume that $a_n, b_n\ge 0$... ? $\endgroup$ – mookid Oct 27 '14 at 20:40
  • 1
    $\begingroup$ The question doesn’t state that $a_n,b_n\ge 0$. Without this it is clearly false by taking both to be the same conditionally-convergent series. $\endgroup$ – Erick Wong Mar 12 '18 at 17:25

Hint: $$ \sqrt {AB}\le A+B $$ because of $(\sqrt A - \sqrt B)^2 \ge 0$.

Note: Similar can be shown to hold true for $A_1, ..., A_n$ and $A_1, A_2, ...$

  • $\begingroup$ We assume that it can be shown by induction that similar holds true for both $A_1,A_2,...,A_n$ and $A_1, A_2,...$ ? $\endgroup$ – BCLC Mar 8 '18 at 16:16
  • $\begingroup$ we can do it :) $\endgroup$ – mookid Mar 9 '18 at 21:43
  • $\begingroup$ mookid, the point is the extension to $\infty$ is true but not trivial... $\endgroup$ – BCLC Mar 12 '18 at 17:17
  • $\begingroup$ this is a special case of Jensen's inequality. $\endgroup$ – mookid Mar 13 '18 at 17:19

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