You'll notice that while the bonus 1000 mana, indeed, doesn't stack; the flat mana it gives AND the conversion of 3% of your mana to ap does stack.

So while it does end up being a bit of a waste if mana is what you're looking for, it does give more ap, in the end, than stacking deathcaps.

You would really have 3000 less mana (the wasted bonus mana from the unique passive) putting you at 3840. 3% of 3840 = 115.2 Then you add the flat ap the item gives to give you a grand total of 160.2 ap per staff. Putting your total ap at:
((160.2*4)+140)*1.3=1015.04
And that's not including the ap gained from masteries, runes, lasthits with Q or kills. All of which would give you more ap.

This build wouldn't work because the permanent mana creation effect is a Unique Passive, so multiple Archangel's Staff doesn't stack. It says that in the end I would have 6840 mana, but that would be impossible since Tear of the Goddess, Manamune, and Archangel's Staff doesn't stack with each other.