Math Help - converge and diverge problem

Does this problem converge absolutely, conditionally, or does it diverge?

the equation: [url=http://img219.imageshack.us/img219/5968/untitled7gf.jpg[/url]

also, the hint is to first show that ln(1 + x) <= x if x > 0

Thanks for any help.

Originally Posted by rcmango

Does this problem converge absolutely, conditionally, or does it diverge?

the equation: [url=http://img219.imageshack.us/img219/5968/untitled7gf.jpg[/url]

also, the hint is to first show that ln(1 + x) <= x if x > 0

Thanks for any help.

As the term inside the log is monotonicaly decreasing and goes to 1 as n
goes to infinity, the log term is monotonicaly decreasing and goes to zero,
thus by the alternating series test the series converges. See the other
for details or this link.

The hint tells us that ln(1+1/n^2)<1/n^2, and as sum(1/n^2, n=1,..., infty)
does converge the series is also absolutly convergent.

RonL