Determine whether the following series is convergent (hint: use the comparison test)

1-(1/2)+(2/3)-(1/3)+(2/4)-(1/4)+....

ie.

(2/2)-(1/2)+(2/3)-(1/3)+(2/4)-(1/3)+...

Am I allowed to turn this into

(1/2)+(1/2)+(1/4)+..... by summing each pair of terms in the series, or is that only allowed if I know the series converges absolutely?

The second option (i.e., you can't arbitrarily group terms in a general series), but even then I can't see how woould you get what you wrote if you group in pairs...it should be, imo, Tonio
If I can do that, then the series is infinitesum{ 1/(n+1) } which equals infinitesum{ 1/n } -1 and since infinitesum{1/n} diverges, so does infinitesum{ 1/n } -1 I would think.

But I didn't use comparison...