does the infinite series $1 - 1/2 + 2/3 -1/3 + 2/4 - 1/4 + 2/5 -1/5...$ converge? does the infinite series  $1 - 1/2 + 2/3 -1/3 + 2/4 - 1/4 + 2/5 -1/5...$ converge? 

I am saying no since it is not nondecreasing. Is this right or should something more be done?
Except for the missing first term, this sequence is equal to the harmonic series, which is divergent. Just group two adjacent terms and you'll see. Making it alternating by splitting the terms doesn't make it more convergent (it rather makes it worse).
1-1/2+2/3-1/3... is converging because if you combine like terms (ie 1-1/2 2/3-1/3 and so on) you get the harmonic series (ie 1+1/2+1/3...) which diverges. Thus 1-1/2+2/3-1/3... diverges.