Root and Ratio tests Inconclusive

[mattmns]

Here is the question in the book:
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Give an example of a divergent infinite series of positive numbers a_{n} such that \lim_{n\rightarrow \infty}a_{n+1}/a_{n} = \lim_{n\rightarrow \infty}a_{n}^{1/n} = 1 and an example of a convergent infinite series of positive numbers with the same property.
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For the divergent series I gave a somewhat trivial example. The series: 1+1+1+1+1+... which is certainly divergent and has the necessary limit properties.

For the convergent series though I am a little clueless. What should I be looking for. I know that the nth root of the "last" term should be 1, and also the "last" term should be approaching 0, but I just can't seem to get anything to satisfy both. Please don't post such a series, but instead if you could give some hints as to how I should go about finding one, and maybe other things that I should know when looking for such a sequence. Thanks!

 

[quasar987]

hint: n^(1/n)-->1

 

[mattmns]

\sum_{n=0}^{\infty} 1/n^2 converges and has the properties. Thanks.

edit... Using your hint again, I could use \sum_{n=0}^{\infty} 1/n as the divergent series with the desired properties which is probably a little nicer than my lame example

 

[quasar987]

Nice.

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