series and limsups

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aguacate
Posts: 209
Joined: Fri Feb 16, 2007 10:29 pm UTC

series and limsups

Postby aguacate » Tue Nov 06, 2007 7:47 pm UTC

The official ratio and root tests use the limsup of an and not just the lim. On the other hand if lim an converges, then limsup=liminf=lim. But then if a series converges, then lim an must not only converge, but converge to 0. So, why limsup?

...

OK, actually I just realized while typing this that its not the limsup of an, but the limsup of (|an|)1/n, for the root test, and likewise for the ratio test. Does anyone have an example where lim(|an|)1/n doesn't converge but lim an converges to 0?
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Owehn
Posts: 479
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Re: series and limsups

Postby Owehn » Tue Nov 06, 2007 7:57 pm UTC

How about letting an be 1/n if n is even, and 0 if n is odd. Then an converges to 0, but |an|1/n approaches 1 on the subsequence of even indices and (equals) 0 on the subsequence of odd indices. That's the best sort of nonconvergence you'll get (though you can probably think of more pathological examples): since |an| approaches zero it is eventually less than 1, so the sequence |an|1/n is eventually bounded between 0 and 1, and hence has convergent subsequences (and in particular, its limsup will be between 0 and 1 as well).
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