Divergence of a series

Face with the series;

Sum from n=1 to infinity of: n/(1+n),

And i'm asked to use any relevant tests and then determine if it converges or not.

I used the ratio test and got it to =1 therefore inconclusive, root test doesnt seem applicable, can't use integral test as the function is not decreasing, is all that correct and is what i've done below correct,

lim (as n --> inf) (n/(1+n)) is 1 as co-efficients of two greatest growing terms are 1/1, as limit doesn't equal 0 the series is divergent.

I'm really unsure with this stuff applying the tests seems simple but the text i have is very shady on how to otherwise deal with the series.

Any help would be GREATLY appreciated.

Cheers.

Yes, that is correct. The terms must go to zero in order for a series to converge. You should try this test first when faced with a series.