Divergence of series - Counter example

Question

I am attempting to disprove following statement with counter example.

If $\sum_{n=1}^{\infty} a_{n}$ converges and lim $b_{n}$ = 0, then $\sum_{n=1}^{\infty} a_{n}b_{n}$ converges

My work: If a series converges then lim $ a_{n} $ = 0. All of the examples I am trying to construct are turning out to be convergent. Can I get some help here?

Hint: if $a_n$ was always positive (or converged absolutely) you could prove the property to be true fairly easily... so you will need to consider $a_n$ to be a convergent series that is not absolutely convergent; think of an alternating series maybe. — User8128, Nov 02 '20 at 03:51

score 1 · Answer 1 · answered Nov 02 '20 at 03:59

1

[Counter-Example] : $a_{n}=b_{n}=\frac{(-1)^n}{\sqrt{n}}$

answered Nov 02 '20 at 03:59

$\sum a_{n}$ is convergent series? – weedfarmer Nov 02 '20 at 04:10
Hello, yes it is by the Alternate Series Test but their product gives you $1/n$ which diverges. – Nov 02 '20 at 04:11

Divergence of series - Counter example

1 Answers1