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?

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

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**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?

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

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

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

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.

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.

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**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.

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

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**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)