# Example of sequence that neither converges nor diverges [closed]

let $a_n$ = sequence that diverge to infinity and $b_n$ = null sequence Then find a sequnce such that $a_nb_n$ neither converges nor diverges

I was thinking $\frac {sin(n)}{n}$

## closed as unclear what you're asking by Winther, mathguy, Marios Gretsas, Jack, NamasteNov 28 '17 at 0:19

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• what do you mean null sequence? – Marios Gretsas Nov 27 '17 at 20:45
• If a sequence doesn't converge then it diverges – David Bowman Nov 27 '17 at 20:47
• @DavidBowman Depending on the author, a sequence that oscillates might be considered neither convergent nor divergent, e.g. $\{(-1)^n\}_{n\in\mathbb{N}}$. – Xander Henderson Nov 27 '17 at 20:49
• @mathguy: denoting every non-convergent sequence as divergent is not widespread at all, it is just very common in the US. To make a distinction between "divergent, unbounded sequences" and "divergent, bounded sequences" is important, both in number theory and (harmonic) analysis, in order to apply summation/integration by parts or (Borel, zeta) regularization techniques. I would not dare to call the vast majority of Italian authors idiots. – Jack D'Aurizio Nov 27 '17 at 21:22
• @mathguy: I can cite Prodi, Giusti, Spagnolo, Modica, Buttazzo and many others. In Italy we usually distinguish between positively divergent and negatively divergent sequences, converging sequences and not-converging sequences. About the partial sums of a sequence, the meaning is pretty obvious: the partial sums of $\{a_n\}_{n\geq 0}$ are $a_0,a_0+a_1,a_0+a_1+a_2,\ldots$. – Jack D'Aurizio Nov 27 '17 at 21:36

## 2 Answers

The product of two sequences that each converge to zero will also converge to zero, so you won't find one.

Added: now that $a_n$ is supposed to diverge to infinity, you should be able to make $a_nb_n$ a constant sequence. Now just multiply every other $b_n$ by $-1$ to make it wobble so much it doesn't converge.

• One was supposed to diverge to infinity sorry – Sarah.B Nov 27 '17 at 21:13
• can you please provide an example? – Sarah.B Nov 27 '17 at 21:28
• A nice series that diverges is $a_n=n$. Can you start with that? – Ross Millikan Nov 28 '17 at 1:20

I think I finally understand what you mean. One sequence converges to 0, the other diverges to $+\infty$, the product does not converge and it does not diverge to infinity.

Take $a_n = \frac 1 n$ (converges to $0$) and $b_n = n$ if $n$ is odd, $b_n = 2n$ if $n$ is even ($b_n$ diverges to $+\infty$). The product alternates between the values $1$ and $2$ forever; so it is not convergent, and it does not diverge to infinity; in fact, it is bounded.

• Thanks. I didn't word the question very well! How about if $a_n$ >0 – Sarah.B Nov 29 '17 at 15:30
• @Sarah.B - I am not sure what you mean; in the example I provided, you already have $a_n>0$ for all $n$. What else do you mean by that? – mathguy Nov 29 '17 at 15:48
• So if both $a_n$ $b_n$ converge to infintiy. – Sarah.B Nov 29 '17 at 16:03
• "Converge to infinity" is a contradiction in terms. Please check the meaning of "converge". Which tells me it's impossible to communicate - too many failed attempts already. Peace! – mathguy Nov 29 '17 at 16:16