_{n}and not just the lim. On the other hand if lim a

_{n}converges, then limsup=liminf=lim. But then if a series converges, then lim a

_{n}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 a

_{n}, but the limsup of (|a

_{n}|)

^{1/n}, for the root test, and likewise for the ratio test. Does anyone have an example where lim(|a

_{n}|)

^{1/n}doesn't converge but lim a

_{n}converges to 0?