$\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely? [duplicate]

Suppose $\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely?

I know that $\sum|a_n|$ does not converge while $\sum a_n$ does and that $\sum|b_n|$ does converge, and also that $\lim_{n \to \infty} |a_nb_n|=0$ but I'm not sure how to proceed from here.

marked as duplicate by Micah, Robert Israel sequences-and-series StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 8 '18 at 1:41

$|a_{n}|<1$ eventually and so $|a_{n}b_{n}|\leq|b_{n}|$ eventually, but $\displaystyle\sum|b_{n}|<\infty$, then so is $\displaystyle\sum|a_{n}b_{n}|<\infty$.
• how when |$a_n$| diverges – user524644 Mar 8 '18 at 1:27
• I don't use that. Rather, if $\displaystyle\sum a_{n}$ exists, then its tail goes to zero. – user284331 Mar 8 '18 at 1:27
• ok but what so what if $\sum |a_nb_n| < \infty$ – user524644 Mar 8 '18 at 1:29