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a) Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely.

I wanted to use the comparison test to show it's true, but I think I got something mixed up. Here's what I have so far.

$b_n$ is bounded $\Rightarrow \exists c > 0: |b_k| < c, \forall k \in \mathbb N$

$\sum a_nb_n < \sum a_nc < c\sum a_n$. I'm very tempted to use the comparison test here and say, as $\sum a_n$ converges absolutetly, then so does $c\sum a_n$, but for that I needed the inverted relation, right? I would need $\sum a_n > c\sum a_n$, which is not true. However isn't it obvious that something absolutely convergent multiplied by a constant is also absolutetly convergent? Is there a mathematical way to write this? Thanks a lot in advance.

b) Refute with a counter example: if $\sum a_n$ converges and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges.

Is this even possible? I mean, it has to be, but it doesn't make sense to me. If something converges, it means it's bounded, right? And I thought, well, since bounded + bounded = bounded, then bounded*bounded would also get me something bounded again.

Anyway, my idea would be to use for $a_n$ an alternating series that ist only convergent, for example $(-1)^n \frac 1 {n^2}$. But something tells me I'm trying to prove $a_n b_n$ is not absolutely convergent.

Thanks a lot in advance guys!

Martin Sleziak
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Clash
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1 Answers1

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Recall that if $\sum a_n$ is finite then $c\cdot\sum a_n = \sum ca_n$. This is true even if the series is not absolutely convergent.

The reason is simple, $\displaystyle\sum a_n = \lim_{k\to\infty}\sum_{n<k} a_n$, and when multiplying by a constant it can jump into the limit.

Now your reasoning is true. If $b_n\le c$ then $\sum a_nb_n\le \sum ca_n\le c\sum a_n$, and the latter converges.

For the second question, in the first one the assumption was the series is absolutely convergent. Take a sequence which is not of this form.

For example $a_n = \dfrac{(-1)^n}{n}$, and $b_n=(-1)^n$. Now what is $\sum a_nb_n$?

Asaf Karagila
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  • Thanks for the swift reply asaf! Quick question, my reasoning did not use the "comparison test", right? – Clash Nov 24 '11 at 17:09
  • @Clash: Yes, but the reasoning that we can replace $b_n$ by their upper bound. – Asaf Karagila Nov 24 '11 at 17:14
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    and that multiplying something absolutely convergent by a constant does not change it's "status", right? thanks again, i love u – Clash Nov 24 '11 at 17:17
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    @Clash: You might want to prove that. It follows from $\sum |ca_n| = \sum |c||a_n| = |c|\sum |a_n|<\infty$. – Asaf Karagila Nov 24 '11 at 17:26
  • perfect, thank you so much! – Clash Nov 24 '11 at 17:30
  • @AsafKaragila what about Cauchy sequence being absolutely convergent? $ |a_n b_n| $ is only absolutely convergent if $b_n$ is abs convergent ?. – Gaurang Mar 09 '21 at 01:43
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    @GaurangChauhan: a series is usually classified as "absolutely convergent" when the series whose terms are the absolute values of the given series converges. "Absolutely convergent" is not usually applied to sequences. – robjohn Mar 09 '21 at 02:59
  • @robjohn thanks for the response. what you wrote makes sense. But as mentioned by AsafKaragila what if $b_n = (-1)^n$. in that case is the series $∑a_nb_n$ abs convergent ? . suppose $b_1 = -1$ and $a_1 = a$ a positive number. $a_1b_1 = -a$. which is not positive. what do you think about this? – Gaurang Mar 09 '21 at 04:43
  • @GaurangChauhan: The premise is that $\sum\limits_{k=0}^\infty a_n$ converges absolutely. If $a_n=a$, the series is not convergent (unless $a=0$). – robjohn Mar 09 '21 at 07:48