# Are the sum and/or product of two divergent sequences divergent?

• Jun 28th 2011, 09:37 PM
CountingPenguins
Are the sum and/or product of two divergent sequences divergent?
As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
• Jun 28th 2011, 10:06 PM
Also sprach Zarathustra
Re: Are the sum and/or product of two divergent sequences divergent?
Quote:

Originally Posted by CountingPenguins
As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?

$\displaystyle S_1=1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}$

$\displaystyle S_2=-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{ 2n})$

$\displaystyle \lim_{n\to\infty}(S_1+S_2)=\ln2$
• Jun 28th 2011, 10:16 PM
Also sprach Zarathustra
Re: Are the sum and/or product of two divergent sequences divergent?
Quote:

Originally Posted by CountingPenguins
As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?

EDIT: Oooops... you asked for sequences and I give you a series...(But it can work as well)

$\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}$ is converges.

But, $\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}\cdot\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}$ isn't.
• Jun 29th 2011, 12:50 AM
emakarov
Re: Are the sum and/or product of two divergent sequences divergent?
Edit: disregard this. The left forum sidebar protruded over the formula for $\displaystyle S_2$ and obscured the minus.

Quote:

Originally Posted by Also sprach Zarathustra
$\displaystyle S_1=\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}$

$\displaystyle S_2=-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{ 2n})$

$\displaystyle \lim_{n\to\infty}(S_1+S_2)=\ln2$

Isn't $\displaystyle S_1+S_2$ the harmonic series? Then it does not converge.
• Jun 29th 2011, 02:07 AM
Also sprach Zarathustra
Re: Are the sum and/or product of two divergent sequences divergent?
Quote:

Originally Posted by emakarov
Edit: disregard this. The left forum sidebar protruded over the formula for $\displaystyle S_2$ and obscured the minus.

Isn't $\displaystyle S_1+S_2$ the harmonic series? Then it does not converge.

Actually I forgot '1' in the sum S_1... (I'll fix it in my post)