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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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 William Elliot Posts: 2,634 Registered: 1/8/12
Re: Can two series, both diverges, multiplied give a series that
converges?

Posted: Oct 6, 2017 11:24 PM

On Fri, 6 Oct 2017, konyberg wrote:

> 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?

s.t = sum(n=1,oo) 1/n(n+1)

Since for all n in N, 0 < 1/n(n+1) < 1/n^2
and sum(n=1,oo) 1/n^2 converges, s.t converges by comparison theorem.

sum(n=1,oo) 1/n^2 converges by integeral test.