- #1

mattmns

- 1,128

- 6

Here is the question in the book:

--------------

Give an example of a divergent infinite series of positive numbers [itex]a_{n}[/itex] such that [itex]\lim_{n\rightarrow \infty}a_{n+1}/a_{n} = \lim_{n\rightarrow \infty}a_{n}^{1/n} = 1[/itex] and an example of a convergent infinite series of positive numbers with the same property.

-------------

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!

--------------

Give an example of a divergent infinite series of positive numbers [itex]a_{n}[/itex] such that [itex]\lim_{n\rightarrow \infty}a_{n+1}/a_{n} = \lim_{n\rightarrow \infty}a_{n}^{1/n} = 1[/itex] and an example of a convergent infinite series of positive numbers with the same property.

-------------

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!

Last edited: