0

Iv'e got a question and I find it difficult to me.

Let $ a_n,b_n$ be sequences. Assume that $a_n+b_n$ converges.

Is $a_n\cdot{b_n}$ essentially converges?

Prove.

Thank you!

Galc127
  • 4,451

1 Answers1

4

No. Consider $a_n = n$, $b_n = -n$.

Umberto P.
  • 52,165
  • I thought about it, but isn't $-n^2$ converges to $-\infty$? – Galc127 Nov 25 '13 at 18:05
  • 1
    Many people (including me) would say $a_n b_n$ diverges. If you want a different example let $a_n = n$ for $n$ even, $a_n = 0$ for $n$ odd, and $b_n = -a_n$. – Umberto P. Nov 25 '13 at 18:14
  • Like you, I think that "converges to infinity" is like "diverges", but we learned different, so I have to use these definitions.

    Your second example is great!

    I wrote $a_n={ n \ when \ n=2m \ 0 \ when \ n=2m-1 $ and $b_n=-a_n$, thus $a_n\cdot{b_n}$ has two partial limits ($0,-\infty$), so it diverges.

    Again, thank you!

     – Galc127 Nov 25 '13 at 18:17