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Re: My galaxy note n7000 shut down and won't turn back on
1) After 2 years, unless you've been treating the battery properly (conditioning it before first use, charging at the 40%-60% discharge level most of the time, etc.), the battery has probably reached the end of its life. (250-500 charge cycles is about normal, and if you only got 250 cycles you can't complain.)
2) Charging by USB is a slow charge, about 100mA at best, so it would take about 36 hours to fully charge a fully discharged battery. (Charging runs about 60% efficiency.) Fully charging a battery that's almost shot? Maybe a week to get at least enough charge to turn the phone on.
Put the supposedly new fully charged battery into the phone and charge it for 24 hours, using a known good stock Note (or newer Samsung) charger and cable. Then if it still doesn't turn on, you've eliminated the battery as a cause.
The charging circuit in the phone could be bad, the motherboard could be bad, the charger or cable you're using could have gone bad. But 2 years out, with the battery probably treated the way I've seen most batteries treated (used down to 10% charge or less before charging), I'd suspect that the original battery is little more than toxic waste at this point. (Lithium is toxic - don't throw it out in the trash.) And the "new fully charged battery" may be "new, fully charged, sitting on the shelf for 2 years" - which means it needs to be conditioned (fully charged until the phone tells you to disconnect the charger, used until the phone tells you to charge it - 3 full cycles), then used properly. (Assuming it's not a cheap useless battery. Some vendors sell Note batteries for about $5. Burning the $5 bill in the winter is a better idea - at least you'd have warm hands for a few seconds.