heropants:
This is what I originally thought but it seems like all of them that I can find on amazon have an automatic shutoff switch. If the charger doesn't have enough current being drawn from it, it shuts off. Since the servo won't always be running, I'm thinking the little bit the nano draws just to process won't be enough (or at least that's what people are speculating online).
https://www.dorkbotpdx.org/blog/paul/battery_pack_load
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