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Topic: Can two series, both diverges, multiplied give a series that converges?
Replies: 22   Last Post: Oct 7, 2017 12:52 AM

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 Markus Klyver Posts: 730 Registered: 5/26/17
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 10:10 AM

Den fredag 6 oktober 2017 kl. 15:42:22 UTC+2 skrev konyberg:
> Consider these two series.
> s = lim (n=1 to inf) Sum(1/n) and t = lim (n=1 to inf) Sum(1/(1+n)).
> Both series diverges, going to infinity.
> Now if we multiply these, we can argue that every product of the new series is smaller or equal to 1/n^2. So it should converge. Or can we?
> Let us write the first as a series without the sigma and the other with sigma.
> s*t = (1+1/2+1/3+ ...) * t. And since the first from s (1 * t) diverges, how can s*t converge?
>
> KON

I'm not sure if the product will converge absolutely, but if multiplication is done in the naive way it'll converge.