Proof involving series

• December 5th 2009, 08:50 AM
katielaw
Proof involving series
(a) If ∑a_n converges and lim b_n = 0, then ∑(a_n)(b_n) converges.

(b) If ∑b_n converges and lim n-> ∞ (a_n)/(b_n) = 1 then ∑ (a_n) converges.

I'm not totally sure how to do either of these. For (a), my guess is since the lim of b_n exists as a real number, 0, then (a_n)(b_n) are going to converge. Similar for (b).
• December 5th 2009, 09:55 AM
Krizalid
are we assuming that $a_n,b_n>0$ ?
• December 5th 2009, 11:03 AM
katielaw
Yes.
• December 5th 2009, 11:07 AM
Krizalid
ah well, then limit comparison test will do the job.

i'll the first one, then you should be able to do the second one.

since $\sum a_n<\infty,$ we'll apply limit comparison test with $a_n,$ so $\lim_{n\to\infty}\frac{a_nb_n}{a_n}=0,$ so $\sum a_nb_n<\infty.$