Convergence and the Alternating Series test

[deruschi12]

Is it possible for a series to converge without the constraint that a_n+1< or equal to a_n? Can we have a convergent series with only the requirement a_n >0 and the limit as x approaches infinity = 0 (i.e. not a decreasing monotonic series)?

If yes include 3 series which disprove the original conjecture stated, the math that shows these series diverge, how you came up with your 3 series and what makes your series diverge

If no include the series to prove your conjecture, the math that shows your series converges, how you made your series and what makes your series converge

 

[hgfalling]

Is this a homework question? If so, it should be posted in the homework section, and you should show what you have done so far, and people will give you hints or help.